diff --git a/multisite/multisite.Rmd b/multisite/multisite.Rmd
index 8456ced..33353aa 100644
--- a/multisite/multisite.Rmd
+++ b/multisite/multisite.Rmd
@@ -1,14 +1,13 @@
 ---
 bibliography: ./refs.bib
 link-citations: true
-numbersections: true
 colorlinks: true
 secnumdepth: 2
-number-sections: true
+number-sections: true 
 linkReferences: true
 output:
   bookdown::html_document2:
-    number_sections: true
+    number_sections: true 
     toc: true
     theme: journal
     extensions: +implicit_figure
@@ -21,13 +20,11 @@ output:
 <!-- title: "10 Things to Know About Multisite or Block-Randomized Trials" -->
 <!-- author: "Methods Guide Author: Kristen Hunter" -->
 
-# Introduction
-
-## What is a multisite or block randomized trial?
+# 1. What is a multisite or block randomized trial?
 
 A multisite or block-randomized trial is a randomized experiment "in which sample members are randomly assigned to a program or a control group *within* each of a number of sites" (@Raudenbush2015).
 
-For illustration, this guide will focus on multisite educational trials, although multisite trials are not unique to education.
+This guide focuses on multisite educational trials for illustration, although multisite trials are not unique to education.
 Multisite trials are a subset of multilevel randomized controlled trials (RCTs), in which units are nested within hierarchical structures, such as students nested within schools nested within districts.
 This guide uses as an illustrative example the case where each site is a school, although they could also be districts or classrooms; thus the term "site" and "school" are used interchangeably.
 
@@ -60,10 +57,10 @@ Given an estimand, the researchers choose their *estimator* to be the coefficien
 To calculate standard errors, they use Huber-White robust standard errors.
 All these choices result in a point *estimate* (e.g. the program increased reading scores by $5$ points) and a measure of uncertainty (e.g. a standard error of $2$ points).
 
-Next, we'll also need some notation.
+We'll also need some notation.
 This guide follows the Neyman-Rubin potential outcomes notation (@Neyman1923, @Imbens2015).
-The outcomes are $Y_{ij}$ for unit $i$ in site $j$.
-The potential outcomes are $Y_{ij}(1)$: the outcome given active treatment, and $Y_{ij}(0)$: the outcome given control treatment.
+The observed outcomes are $Y_{ij}$ for unit $i$ in site $j$.
+The potential outcomes are $Y_{ij}(1)$, the outcome given active treatment, and $Y_{ij}(0)$, the outcome given control treatment.
 The quantity $B_{ij}$ is the unit-level intention-to-treat effect (ITT) $B_{ij} =  Y_{ij}(1) - Y_{ij}(0)$.
 If there is no noncompliance, the ITT is the ATE, as defined above.
 Then $B_j$ is the average impact at site $j$, $B_j = 1/N_j \sum_{i = 1}^{N_j} B_{ij}$ where $N_j$ is the number of units at site $j$.
@@ -71,17 +68,17 @@ Finally, $N = \sum_{j = 1}^{J} N_j$.
 
 This guide is structured around the choices an analyst must make concerning estimand and estimators, and the resulting consequences.
 The choice of estimand impacts the substantive conclusion that a researcher makes.
-The choice of estimator and standard error estimator results in different statistical properties, including a potential tradeoff between bias and variance.
+The choice of estimator and standard error estimator results in different statistical properties, including a potential trade off between bias and variance.
 This guide summarizes material using the framework provided by @Miratrix2020.
 
-# A multisite trial is a type of a blocked or stratified randomized experiment.
+# 2. A multisite trial is a type of a blocked or stratified randomized experiment.
 
 ## A multisite trial is fundamentally a blocked or stratified RCT.
 
 A multisite trial is a blocked RCT with 2 levels: randomization occurs at the student level (level 1) within blocks defined by sites/schools (level 2).
 For example, in a study of a new online math tool for high school students, randomization occurs at the student level within blocks defined by sites/schools.
 Perhaps half of students at each school are assigned to the status quo / control treatment (no additional math practice),
-and half are assigned to theactive treatment (an offer of additional math practice at home using an online tool).
+and half are assigned to the active treatment (an offer of additional math practice at home using an online tool).
 
 Because of the direct correspondence between multisite trials and blocked experiments, statistical properties of blocked experiments also translate directly to multisite experiments.
 The main difference between a traditional blocked RCT and a multisite experiment is that in many blocked RCTs, the researcher is able to choose the blocks.
@@ -92,24 +89,23 @@ Multisite experiments have structural blocks, such as districts, schools, or cla
 The type of block can impact variance estimation, as shown in @Pashley2021 and @Pashley2022.
 
 The [EGAP Metaketa Projects](https://egap.org/our-work-0/the-metaketa-initiative/){target="_blank"}
-are also multisite trials: the 5--7 countries that contain sites for each study are fixed and chosen in advance by the different research teams.
+are also multisite trials: the 5 to 7 countries that contain sites for each study are fixed and chosen in advance by the different research teams.
 
 ## A multisite trial is not a cluster-randomized trial
 
 A different type of RCT is a [cluster-randomized
-design](https://egap.org/resource/10-things-to-know-about-cluster-randomization/){target="_blank"}
-,
+design](https://egap.org/resource/10-things-to-know-about-cluster-randomization/){target="_blank"},
 in which entire schools are assigned to either the active treatment or control
 treatment.
 [This video explains the difference between cluster and
 block-randomized designs](https://youtu.be/bL2U9z8hX1k){target="_blank"}.
-In a multisite trial trial, treatment is assigned **within a block to individual units**.
+In a multisite trial, treatment is assigned **within a block to individual units**.
 In a cluster-randomized trial, treatment is assigned to **groups** of units.
 Some designs [combine cluster- and
-block-randomization](https://declaredesign.org/r/designlibrary/reference/block_cluster_two_arm_designer.html){target="_blank"}
-.
+block-randomization](https://declaredesign.org/r/designlibrary/reference/block_cluster_two_arm_designer.html){target="_blank"}.
 
-Another design that is not a multisite or block-randomized trial is an experiment that takes place in only one school and assigns individual students to active treatment and control treatment: this study has only one site and thus differences
+Another design that is not a multisite or block-randomized trial is an experiment that takes place in only one school and assigns individual students to active treatment and control treatment.
+This type of study has only one site and thus differences
 between sites do not matter in this design.
 
 ## Why choose a multisite or block-randomized trial design?
@@ -118,9 +114,9 @@ In most contexts, blocking reduces estimation error over an unblocked (completel
 Thus, blocked experiments generally offer higher statistical power than unblocked experiments.
 Blocking is most helpful in increasing precision and statistical power in the setting where there is variation in the outcome, and where the blocks are related to this variation.
 
-In multisite trials as compared to block-randomized trials, the researcher cannot purposely construct blocks to reduce variation, because they are defined by pre-existing sites.
+In multisite trials as compared to block-randomized trials, the researcher typically cannot purposely construct blocks to reduce variation, because they are defined by pre-existing sites.
 However, the researcher can hope, and often expect, that sites naturally explain some between-site variation.
-For example, if some schools tend to have larger impacts than others, and the size of the impact is related to the average income of families attending that school, then blocked randomization using the school as a block improves efficiency over complete randomization.
+For example, if some schools tend to have higher outcomes than others, then blocked randomization using the school as a block improves efficiency over complete randomization.
 
 Randomizing with purposefully created blocks or pre-existing sites also helps analysts learn about how treatment effects may vary across the sites or groups of people categorized into the blocks.
 If a new treatment should help the lowest performing students most, but in any given study most students are not the lowest performing, then researchers may prefer to create blocks of students within schools with the students divided by their previous performance.
@@ -132,9 +128,9 @@ Often, in a multisite trial with treatment administered by site administrators (
 In other studies, the construction and choice of blocking criteria is a choice.
 @Pashley2022 shows that blocking is generally beneficial, but also explores settings in which it may be harmful.
 Blocking does result in fewer degrees of freedom, but in practice this reduction is rarely an issue, unless an experiment is very small [@Imai2008].
-Any use of blocking requires that an analyst keep track of the blocks and also that an analyst reflect the blocks in subsequent analysis: in many circumstances estimating average treatment effects from a block-randomized experiment while ignoring the blocks will yield biased estimates of the underlying targeted estimands (See ["The trouble with 'controlling for blocks'"](https://declaredesign.org/blog/biased-fixed-effects.html)  and ["Estimating Average Treatment Effects in Block Randomized Experiments"](https://egap.org/resource/sd-block-rand) for demostrations of bias arising from different approaches to weighting by blocks.)
+Any use of blocking requires that an analyst keep track of the blocks and also that an analyst reflect the blocks in subsequent analysis: in many circumstances estimating average treatment effects from a block-randomized experiment while ignoring the blocks will yield biased estimates of the underlying targeted estimands (see ["The trouble with 'controlling for blocks'"](https://declaredesign.org/blog/biased-fixed-effects.html)  and ["Estimating Average Treatment Effects in Block Randomized Experiments"](https://egap.org/resource/sd-block-rand) for demonstrations of bias arising from different approaches to weighting by blocks).
 
-# Analysis can either target the population in the experiment, or a broader population.
+# 3. Analysis can either target the population in the experiment, or a broader population.
 
 The first choice a researcher must make in defining their estimand is the population of interest.
 The researcher may want to focus on the **finite population**: only those units in the experimental pool or sample.
@@ -166,7 +162,7 @@ Although the point estimates from either perspective will often be the same, the
 
 For more discussion of the consequence of the super population and finite population frameworks, see @Schochet2016 and @Pashley2021.
 
-# The average site effect is not the same as the average person effect.
+# 4. The average site effect is not the same as the average person effect.
 
 The second choice a researcher makes is the target of inference: is the researcher interested in the **average student**, or
 the **average site** (@Miratrix2020)?
@@ -190,10 +186,10 @@ The average site impact is
 \]
 Note that in the case where all sites are of the same size, or all sites have the same impact, then these two estimands are the same.
 
-To summarize the previous two sections, there have been two axes of choices: the population of interest (FP or SP for finite and super population), and the target of inference (persons or sites).
+To summarize, this section and the prior section have given two axes of choices: the population of interest (FP or SP for finite and super population), and the target of inference (persons or sites).
 These choices result in four possible estimands: FP-persons, SP-persons, FP-sites, and SP-sites.
 
-# There are many widely-used estimators that target the same estimands, including design-based, linear regression, and multilevel models.
+# 5. There are many widely-used estimators that target the same estimands, including design-based, linear regression, and multilevel models.
 
 After choosing an *estimand*, the researcher must then choose an *estimator*, a process to arrive at the estimate of interest.
 There are three main categories of estimators: **design based**, **linear regression**, and **multilevel modeling**.
@@ -208,9 +204,10 @@ The different categories of estimator differ both philosophically and practicall
 Each category assumes a different source of randomness, and thus has a different statistical justification.
 
 **Design-based** estimators specifically target the four estimands outlined above.
-The only source of uncertainty is assumed to be the treatment assignment: which units happened to be assigned to the active treatment, and which happened to be assigned to the control treatment.
+The main source of uncertainty is assumed to be the treatment assignment: which units happened to be assigned to the active treatment, and which happened to be assigned to the control treatment.
 This assumption is the reason for their name; the uncertainty in the estimates is by design, from the purposeful randomization of units.
 Using design-based estimators is also sometimes called Neymanian inference, as the estimators and properties were first introduced by Neyman (@Neyman1923).
+Design-based estimators can also incorporate uncertainty from sampling when using a super population framework.
 
 **Linear regression** estimators are the most familiar to many researchers.
 With these estimators, the observed outcomes are assumed to be a linear function of the treatment assignment, (optionally) site-specific effects, (optionally) covariates, and random error.
@@ -232,7 +229,7 @@ For a more comprehensive look at multilevel models, see @Raudenbush2015.
 Let's examine a few popular models among linear regression and multilevel models in more detail.
 Note that these models as presented do not include covariates, but covariates can easily be incorporated to increase power if the analyst is willing to increase bias by a small amount in exchange (often a very small amount if the experiment is large enough) [@lin2013agnostic].
 
-## Linear regression model assumptions
+## Common linear regression models
 
 **Fixed effects with a constant treatment (FE)**
 
@@ -253,13 +250,15 @@ Y_{ij} = \sum_{k = 1}^{J} \alpha_k \text{Site}_{k,ij} +
 \]
 Given a series of site-specific treatment estimates $\hat{\beta}_j$, these estimates are then averaged, with weights by either simple weighting (see @Clark2011) or by site size.
 
-## Multilevel model assumptions
+## Common multilevel models
 
-Once an analyst selects a multilevel model, for site intercepts and site impacts they must decide: what is considered random, and what is considered fixed?
+Once an analyst selects multilevel modeling, for site intercepts and site impacts they must decide: what is considered random, and what is considered fixed?
 
 **Fixed intercept, random treatment coefficient (FIRC)**
 
 This model is similar to the fixed effects models above, but assumes that the site impact $\beta_j$ is drawn from a shared distribution.
+The FIRC model was more recently designed to handle bias issues that arise when the proportion of units treated varies across sites.
+
 \begin{align*}
 \text{Level 1}\qquad & Y_{ij} = \sum_{k = 1}^{J} \alpha_k
 \text{Site}_{k,ij} + \beta_j T_{ij} + e_{ij}\\
@@ -269,7 +268,7 @@ See @Raudenbush2015 and @Bloom2017.
 
 **Random intercept, random treatment coefficient (RIRC)**
 
-This model further generalizes to assume that both the site intercept and site impact are drawn from shared distributions.
+This model is an older version of multilevel models, and assumes that both the site intercept and site impact are drawn from shared distributions.
 \begin{align*}
 \text{Level 1}\qquad & Y_{ij} = A_j + \beta_j T_{ij} + e_{ij}\\
 \text{Level 2}\qquad & \beta_j = \beta + b_j\\
@@ -292,16 +291,15 @@ For example, a fixed-effects model can weigh each person by their inverse chance
 Weighted regression for traditional regression is discussed in @Miratrix2020, and weighted regression for multilevel models is discussed in @Raudenbush2020.
 
 
-# Some estimators attempt to reduce variance by increasing bias.
+# 6. Some estimators attempt to reduce variance by increasing bias.
 
 Each category of estimator (design, regression, and multilevel) results in a different estimation approach.
 One way to characterize the categories is the weights induced by the choice of estimator.
 The properties of each estimator also result in different consequences for bias and variance.
-Design-based estimators are unbiased, but may not always afford the most precise estimates.
+Design-based estimators are generally unbiased, but may not always afford the most precise estimates.
 In general, model-based estimators trade bias for variance.
 Thus, they can sometimes have a lower mean squared error than design-based estimators.
 One way that model-based estimators increase precision is through the easy incorporation of covariates.
-Although design-based estimators can also incorporate covariates, it is not always as straightforward.
 Covariate adjustment methods that incorporate covariates result in the equivalent to a weighted regression approach.
 
 
@@ -314,10 +312,10 @@ Then, the overall estimate is a weighted combination of these estimates, weighte
 The design-based estimators are
 \begin{align*}
 \hat{\beta}_{DB-persons} &= \sum_{j = 1}^{J} \frac{N_j}{N} \hat{B_j} \\
-\hat{\beta}_{DB-sites} &= \sum_{j = 1}^{J} \frac{1}{J} \hat{B_j}
+\hat{\beta}_{DB-sites} &= \sum_{j = 1}^{J} \frac{1}{J} \hat{B_j}.
 \end{align*}
 Design-based estimators are generally *unbiased* for their corresponding estimands (person-weighted or site-weighted).
-Unbiasedness does not hold for one superpopulation model; see @Pashley2022 for more details.
+Unbiasedness does not hold for one super population model; see @Pashley2022 for more details.
 
 
 ## Linear regression estimators
@@ -329,14 +327,13 @@ The estimator is
 \hat{\beta}_{FE} = \sum_{j = 1}^{J} \frac{N_j p_j (1 - p_j)}{Z} \hat{B_j},
 \]
 where $p_j$ is the proportion treated at site $j$.
-The quantity $Z$ is a normalizing constant: $Z = \sum_{j = 1}^{J} N_j p_j (1-p_j)$ to ensure the weights sum to one.
-In this model, the weights include $p_j$, which tells us information about the precision of the estimate for that site:
-$N_j p_j (1 - p_j)$ is the inverse of $Var(\hat{\beta_j})$.
-This expression shows that sites with larger $N_j$, or have $p_j$ closer to $0.5$, have larger weights.
+The quantity $Z$ is a normalizing constant, so $Z$ is defined as $\sum_{j = 1}^{J} N_j p_j (1-p_j)$ to ensure the weights sum to one.
+The weights are $N_j p_j (1 - p_j)$, which is the inverse of $Var(\hat{\beta_j})$, so the weights are related to the precision of the estimate for each site.
+This expression shows that sites with larger $N_j$, or that have $p_j$ closer to $0.5$, have larger weights.
 
 The FE estimator is not generally unbiased for either person-weighted or site-weighted estimands.
 If the impact size $B_j$ is related to the weights ($N_j p_j (1 - p_j)$), then the estimator could be biased.
-For example, if sites that treat a higher proportion of treated units also have a large impact, then $B_j$ can be related to $p_j (1- p_j)$.
+For example, if sites that treat a higher proportion of treated units also experience a larger  treatment impact, then $B_j$ can be related to $p_j (1- p_j)$.
 This setting is plausible for example if sites with more resources to intervene on more students also implement the intervention more effectively.
 If larger sites are more effective, then $B_j$ can be related to $N_j p_j (1- p_j)$.
 
@@ -349,20 +346,22 @@ In contrast, the FE-inter model ends up with weights identical to the design-bas
 ## Multilevel model estimators
 
 Multilevel models also result in precision weighting, but in these models the estimated precision also takes into account the assumed underlying variance in site impacts.
-For example, consider the FIRC model:
+For example, the FIRC model can be expressed roughly as:
 \[
 \hat{\beta}_{ML-FIRC*} = \sum_{j = 1}^{J} \frac{1}{Z}
-\left(\frac{\sigma^2}{N_j p_j ( 1 - p_j)} + \tau^2\right)^{-1}
+\left(\frac{\sigma^2}{N_j p_j ( 1 - p_j)} + \tau^2\right)^{-1} \hat{B_j},
 \]
 where $Z$ is again a normalizing constant, $Z = \sum_{j = 1}^{J} \left(\frac{\sigma^2}{N_j p_j ( 1 - p_j)} + \tau^2\right)^{-1}$.
-This equation assumes that $b_j$ has known variance $\tau^2$, and $e_{ij}$ has known variance $\sigma^2$.
+This equation assumes that the $b_j$ have known variance $\tau^2$, and the $e_{ij}$ have known variance $\sigma^2$.
 In general, we do not know these quantities, and instead must estimate them.
 However, we can see that the implied precision weights incorporate the additional uncertainty assumed in the value of $b_j$.
 
 The RIRC model imposes the same structure on the site impacts, and thus the weights are similar to the FIRC model.
-The RICC model assumes a constant treatment impact, and thus is essentially equivalent to the fixed effects with constant treatment model (FE) when it comes to estimating the site impacts.
+The RICC model assumes a constant treatment impact, and thus is essentially equivalent to the precision-weighted fixed effects with constant treatment model (FE) when it comes to estimating the site impacts.
 
-We summarize the weights below.
+We summarize the weights in the table below.
+The following table includes additional estimators that are not discussed in this guide;
+for more information about these additional estimators, see @Miratrix2020.
 
 | Weight name                       | Weight                          | Estimators |
 | -----                             | -----                           | -----      |
@@ -371,17 +370,17 @@ We summarize the weights below.
 | Random-effect precision-weighting | $w_j \propto \left[\hat{\tau} + N_j p_j (1 - p_j)\right]^{-1}$ \ (approximately) | $\hat{\beta}_{ML-FIRC}$, $\hat{\beta}_{ML-RIRC}$ |
 | Unbiased site-weighting           | $w_j \propto 1$                 | $\hat{\beta}_{DB-FP-site}$, $\hat{\beta}_{DB-SP-site}$, $\hat{\beta}_{FE-weight-site}$, $\hat{\beta}_{FE-inter-site}$ |
 
-# For each estimator that achieves a point estimator, there may be multiple options for estimating standard errors.
+# 7. For each estimator that achieves a point estimator, there may be multiple options for estimating standard errors.
 
 The difference between the finite population and super population framework comes into focus when calculating the standard error of various estimators.
 In general, the super population framework results in larger estimates of error because of the additional uncertainty induced by assuming the sites observed are randomly drawn from a larger population.
-In general, variation can be characterized by either *within site* variation or *between site* variation.
+In general, variation in the control outcome can be broken down into *within site* variation and *between site* variation.
 In the finite population framework, estimators calculate variation *within* sites, and then estimators average this variation across sites.
 In the super population framework, estimators look at the variation *between* sites to "capture both any within-site estimation error along with the uncertainty associated with sampling sites from a larger population" (@Miratrix2020).
 For both approaches, modeling assumptions can stabilize uncertainty estimation procedures, but also risk inducing bias if the modeling assumptions are wrong.
 
 For design-based estimators, for the finite population framework Neyman developed a conservative estimator for the standard error using the observed outcomes.
-First, within-site uncertainty is estimated, and then these estimates are averaged with weights according to the target estimand.
+First, within-site uncertainty is estimated for each site, and then these estimates are averaged with weights according to the target estimand.
 The super population framework induces more complicated expressions that take into account the additional population variance.
 The details of standard errors for super population design-based estimators are beyond the scope of this guide.
 
@@ -392,7 +391,7 @@ assumption (see @Weiss2019 and @Richburg-Hayes2008).
 Robust standard errors fall into a design-based approach instead of a model-based approach [@lin2013agnostic; Chapter 3 of @gerber2012field].
 Huber-White standard errors correspond to the finite population framework, while the asymptotic theory justifying traditional cluster robust standard errors corresponds to the super population framework in regards the clusters. In a [cluster-randomized trial](https://egap.org/resource/10-things-to-know-about-cluster-randomization/), treatment is assigned to clusters, so there is also a finite-population-of-clusters perspective on cluster robust standard errors that is approximated in what are commonly known as CR2 standard errors [@Pustejovsky2018].
 
-To briefly summarize this correspondence, first consider the motivation behind robust standard error estimators.
+To briefly summarize the correspondence between standard error estimators and the assumed population, first consider the motivation behind robust standard error estimators.
 In the FE model, treatment effects are assumed to be constant across sites.
 Thus, if there is truly treatment effect heterogeneity, units in different sites will have different amounts of variation, and this variation will be incorporated into the error term.
 The assumption of $iid$ standard errors will be broken.
@@ -409,15 +408,15 @@ Generally, maximum likelihood theory is applied, which "requires a complete mode
 FIRC and RIRC models naturally produce standard errors under the super population framework, while RICC essentially takes a
 finite population framework because the treatment impacts are not assumed to be drawn from a super population, as they are assumed to be consistent across sites.
 
-# The analyst's choices of estimand, estimator, and standard error estimator matter in some cases, and matter less in others.
+# 8. The analyst's choices of estimand, estimator, and standard error estimator matter in some cases, and matter less in others.
 
 After discussing the different choices a researcher can make in analyzing a multisite trial, a big question remains: how do these choices impact empirical results?
 Which of these choices have a substantial impact on the conclusion we reach, and which do not matter as much?
-@Miratrix2020 conducted an empirical study to investigate these questions using 12 large multisite trials.
+@Miratrix2020 conducted an empirical study to investigate these questions using 12 large multisite trials, backed up by simulation studies in certain cases.
 
 ## Point estimates
 
-First, they consider the impact on point estimates.
+First, they consider the impact of choices on point estimates.
 The authors ask, "to what extent can the choice of estimator of the overall average treatment effect result in a different impact estimate?"
 In general, the authors find that the choice of estimator can substantially impact the point estimates, although the degree of impact depends on the choice.
 The authors reach the following conclusions.
@@ -433,31 +432,30 @@ They found that "the range of estimates across all estimators is rarely meaningf
 
 The unbiased design-based estimator and the precision-weighted fixed effect estimate both target the person-weighted estimand.
 There was little difference in estimates between these estimators.
-Most likely, "this implies that the potential bias in the bias-precision tradeoff to the fixed effect estimators is negligible in practice."
-Other [authors](https://egap.org/resource/sd-block-rand/){target="_blank"} have been able to create situations in which the bias-precision tradeoff is more severe.
+Most likely, "this implies that the potential bias in the bias-precision trade off to the fixed effect estimators is negligible in practice."
+Other [authors](https://egap.org/resource/sd-block-rand/){target="_blank"} have been able to create situations in which the bias-precision trade off is more severe.
 
 **For site-weighted estimands, the choice of estimator can matter.**
 
 FIRC estimates did differ from the unbiased design-based site estimator.
-FIRC can be seen as an adaptive estimator: when there is little observed variation between sites, it tends to be more similar to person-weighted estimate instead of the site-weighted estimate.
+FIRC can be seen as an adaptive estimator: when there is little estimated variation in impacts between sites, it tends to be more similar to the person-weighted estimate instead of the site-weighted estimate.
 
-**Different estimators have different bias-variance tradeoffs.**
+**Different estimators have different bias-variance trade offs.**
 
-Finally, the authors consider the empirical bias-variance tradeoff of different estimators, and find:
+Finally, the authors consider the empirical bias-variance trade off of different estimators, and find:
 
 - FE estimators have little bias, but also do not improve precision much over design-based estimators.
-- To further investigate, they conducted a simulation study and found:
-  - FIRC tends to have lower mean squared error than design-based estimators.
-  - Larger site impact heterogeneity results in more biased estimates for FIRC.
-  - Even with more site impact heterogeneity, the mean squared error for FIRC estimators is still generally lower.
-  - Coverage for design-based estimators is more reliable, especially when site size is variable and site size is correlated with impact.
+- FIRC tends to have lower mean squared error than design-based estimators.
+- Larger site impact heterogeneity results in more biased estimates for FIRC.
+- Even with more site impact heterogeneity, the mean squared error for FIRC estimators is still generally lower.
+- Coverage for design-based estimators is more reliable, especially when site size is variable and site size is correlated with impact.
 
 ## Standard errors
 
 The second question concerns the choice of standard error estimators.
 The authors ask, "to what extent can the choice of estimator of the standard error of the overall average treatment effect result in a different estimated standard error?"
 
-Similar to for point estimates, the choice of standard error estimator can substantially impact the estimated standard error.
+The choice of standard error estimator can substantially impact the estimated standard error.
 The authors reach the following conclusions.
 
 **The choice of estimand impacts the standard error.**
@@ -477,20 +475,19 @@ The authors further conclude that for super population site-weighted estimands,
 Through a simulation study, they find that super population standard errors can underestimate the true error.
 The design-based super population standard error estimator is particularly prone to underestimate the standard error compared to multilevel models, and can be unstable, in that it estimates a wide range of different values across simulations.
 
-# The choice of estimator impacts power.
+# 9. The choice of estimator impacts power.
 
-Given the discussion thus far, it is not surprising that modeling choices made by the analyst also impacts power.
+Given the discussion thus far, it is not surprising that modeling choices made by the analyst also impacts statistical power.
 
-First, we define an important quantity in power calculations: the intraclass correlation coefficient (ICC).
-Broadly, variation can be categorized into *within*-site variation, and *between*-site variation.
-Blocking helps compensate for between-site variation.
-In educational trials, the ICC is the proportion of variance in the outcome that lies *between* sites (@Schochet2016).
+To further understand power, we define an important quantity in power calculations: the intraclass correlation coefficient (ICC).
+Broadly, variation in the observed control outcomes can be categorized into *within*-site variation, and *between*-site variation.
+In educational trials, the ICC is the proportion of variation in the outcome that lies *between* sites (@Schochet2016).
 The ICC is defined as the ratio of the variance at the site level divided by the overall variance of the individual outcomes.
 This quantity plays a different role in block-randomized trial power analysis depending on the target of inference chosen by the analyst.
 ICC is also used in the design and analysis of cluster-randomized trials.
 
 We consider two different estimators and how they impact power.
-First, consider a version of the FE model that has been expanded to include level 1 (student) covariates.
+First, consider a version of the finite population FE model that has been expanded to include level 1 (student) covariates.
 The standard error for the ATE estimator is
 
 \[
@@ -498,15 +495,15 @@ SE = \sqrt{\frac{(1-\text{ICC})(1-R^2_{1})}{\bar{T}(1 - \bar{T}) J \bar{n}}},
 \]
 
 where $ICC$ is the intraclass correlation, $R_1^2$ is the proportion of variation explained by level 1 (student) covariates, $\bar{T}$ is the average number of treated units per site, $J$ is the number of sites, and $\bar{n}$ is the average number of students per site.
-or more information about this standard error expression, see the technical appendix of @Hunter2022.
+For more information about this standard error expression, see the technical appendix of @Hunter2022.
 
-In contrast, consider the RIRC model.
+In contrast, consider the super population RIRC model.
 The standard error for the ATE estimator is
 \[
 SE = \sqrt{\frac{\text{ICC} \omega}{J} + \frac{(1-\text{ICC})(1-R^2_{1})}{\bar{T}(1 - \bar{T}) J \bar{n}}},
 \]
-where $\omega$ is the ratio between the impact variation and the control outcome variation.
-We can see that in doing super population inference, the standard error has an additional term which is nonnegative, so it will be as least as large as the standard error from finite population inference.
+where $\omega$ is the ratio between the cross-site impact variation and the control outcome variation.
+We can see that in doing super population inference, the standard error has an additional term which is non-negative, so it will be as least as large as the standard error from finite population inference.
 A larger standard error will result in lower power.
 
 Examining these standard error formulae also gives a better understanding of what factors impact power.
@@ -520,7 +517,7 @@ The package also calculates sample size requirements and minimum detectable effe
 The newly-developed PUMP package (@Hunter2022) extends the functionality of PowerUpR! to experiments with multiple outcomes, in addition to providing user-friendly tools for exploring the sensitivity of power to different assumptions.
 
 
-# Takeaway advice for researchers on multisite trials
+# 10. Takeaway advice for researchers on multisite trials
 
 Many research plans and analyses do not clearly specify an estimand.
 This lack of clarity can both obscure the goal, and result in poor analysis choices.
diff --git a/multisite/multisite.html b/multisite/multisite.html
index 629119d..7588f21 100644
--- a/multisite/multisite.html
+++ b/multisite/multisite.html
@@ -13,6 +13,19 @@
 
 <title>multisite.knit</title>
 
+<script>// Pandoc 2.9 adds attributes on both header and div. We remove the former (to
+// be compatible with the behavior of Pandoc < 2.8).
+document.addEventListener('DOMContentLoaded', function(e) {
+  var hs = document.querySelectorAll("div.section[class*='level'] > :first-child");
+  var i, h, a;
+  for (i = 0; i < hs.length; i++) {
+    h = hs[i];
+    if (!/^h[1-6]$/i.test(h.tagName)) continue;  // it should be a header h1-h6
+    a = h.attributes;
+    while (a.length > 0) h.removeAttribute(a[0].name);
+  }
+});
+</script>
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-        .attr('data-bs-target', '#' + id)   // BS5
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@@ -324,7 +335,6 @@
 
 
 
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 div.csl-bib-body { }
@@ -372,9 +382,6 @@
 summary {
   display: list-item;
 }
-details > summary > p:only-child {
-  display: inline;
-}
 pre code {
   padding: 0;
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@@ -459,7 +466,7 @@
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-<button type="button" class="btn btn-default btn-xs btn-secondary btn-sm dropdown-toggle" data-toggle="dropdown" data-bs-toggle="dropdown" aria-haspopup="true" aria-expanded="false"><span>Code</span> <span class="caret"></span></button>
+<button type="button" class="btn btn-default btn-xs btn-secondary btn-sm dropdown-toggle" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false"><span>Code</span> <span class="caret"></span></button>
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 <li><a id="rmd-show-all-code" href="#">Show All Code</a></li>
 <li><a id="rmd-hide-all-code" href="#">Hide All Code</a></li>
@@ -473,60 +480,56 @@
 
 <div id="TOC">
 <ul>
-<li><a href="#introduction" id="toc-introduction"><span class="toc-section-number">1</span> Introduction</a>
+<li><a href="#what-is-a-multisite-or-block-randomized-trial" id="toc-what-is-a-multisite-or-block-randomized-trial"><span class="toc-section-number">1</span> 1. What is a multisite or block randomized trial?</a>
 <ul>
-<li><a href="#what-is-a-multisite-or-block-randomized-trial" id="toc-what-is-a-multisite-or-block-randomized-trial"><span class="toc-section-number">1.1</span> What is a multisite or block randomized trial?</a></li>
-<li><a href="#preliminaries-estimands-estimators-and-estimates" id="toc-preliminaries-estimands-estimators-and-estimates"><span class="toc-section-number">1.2</span> Preliminaries: estimands, estimators, and estimates</a></li>
+<li><a href="#preliminaries-estimands-estimators-and-estimates" id="toc-preliminaries-estimands-estimators-and-estimates"><span class="toc-section-number">1.1</span> Preliminaries: estimands, estimators, and estimates</a></li>
 </ul></li>
-<li><a href="#a-multisite-trial-is-a-type-of-a-blocked-or-stratified-randomized-experiment." id="toc-a-multisite-trial-is-a-type-of-a-blocked-or-stratified-randomized-experiment."><span class="toc-section-number">2</span> A multisite trial is a type of a blocked or stratified randomized experiment.</a>
+<li><a href="#a-multisite-trial-is-a-type-of-a-blocked-or-stratified-randomized-experiment." id="toc-a-multisite-trial-is-a-type-of-a-blocked-or-stratified-randomized-experiment."><span class="toc-section-number">2</span> 2. A multisite trial is a type of a blocked or stratified randomized experiment.</a>
 <ul>
 <li><a href="#a-multisite-trial-is-fundamentally-a-blocked-or-stratified-rct." id="toc-a-multisite-trial-is-fundamentally-a-blocked-or-stratified-rct."><span class="toc-section-number">2.1</span> A multisite trial is fundamentally a blocked or stratified RCT.</a></li>
 <li><a href="#a-multisite-trial-is-not-a-cluster-randomized-trial" id="toc-a-multisite-trial-is-not-a-cluster-randomized-trial"><span class="toc-section-number">2.2</span> A multisite trial is not a cluster-randomized trial</a></li>
 <li><a href="#why-choose-a-multisite-or-block-randomized-trial-design" id="toc-why-choose-a-multisite-or-block-randomized-trial-design"><span class="toc-section-number">2.3</span> Why choose a multisite or block-randomized trial design?</a></li>
 <li><a href="#why-not-block" id="toc-why-not-block"><span class="toc-section-number">2.4</span> Why not block?</a></li>
 </ul></li>
-<li><a href="#analysis-can-either-target-the-population-in-the-experiment-or-a-broader-population." id="toc-analysis-can-either-target-the-population-in-the-experiment-or-a-broader-population."><span class="toc-section-number">3</span> Analysis can either target the population in the experiment, or a broader population.</a></li>
-<li><a href="#the-average-site-effect-is-not-the-same-as-the-average-person-effect." id="toc-the-average-site-effect-is-not-the-same-as-the-average-person-effect."><span class="toc-section-number">4</span> The average site effect is not the same as the average person effect.</a></li>
-<li><a href="#there-are-many-widely-used-estimators-that-target-the-same-estimands-including-design-based-linear-regression-and-multilevel-models." id="toc-there-are-many-widely-used-estimators-that-target-the-same-estimands-including-design-based-linear-regression-and-multilevel-models."><span class="toc-section-number">5</span> There are many widely-used estimators that target the same estimands, including design-based, linear regression, and multilevel models.</a>
+<li><a href="#analysis-can-either-target-the-population-in-the-experiment-or-a-broader-population." id="toc-analysis-can-either-target-the-population-in-the-experiment-or-a-broader-population."><span class="toc-section-number">3</span> 3. Analysis can either target the population in the experiment, or a broader population.</a></li>
+<li><a href="#the-average-site-effect-is-not-the-same-as-the-average-person-effect." id="toc-the-average-site-effect-is-not-the-same-as-the-average-person-effect."><span class="toc-section-number">4</span> 4. The average site effect is not the same as the average person effect.</a></li>
+<li><a href="#there-are-many-widely-used-estimators-that-target-the-same-estimands-including-design-based-linear-regression-and-multilevel-models." id="toc-there-are-many-widely-used-estimators-that-target-the-same-estimands-including-design-based-linear-regression-and-multilevel-models."><span class="toc-section-number">5</span> 5. There are many widely-used estimators that target the same estimands, including design-based, linear regression, and multilevel models.</a>
 <ul>
-<li><a href="#linear-regression-model-assumptions" id="toc-linear-regression-model-assumptions"><span class="toc-section-number">5.1</span> Linear regression model assumptions</a></li>
-<li><a href="#multilevel-model-assumptions" id="toc-multilevel-model-assumptions"><span class="toc-section-number">5.2</span> Multilevel model assumptions</a></li>
+<li><a href="#common-linear-regression-models" id="toc-common-linear-regression-models"><span class="toc-section-number">5.1</span> Common linear regression models</a></li>
+<li><a href="#common-multilevel-models" id="toc-common-multilevel-models"><span class="toc-section-number">5.2</span> Common multilevel models</a></li>
 </ul></li>
-<li><a href="#some-estimators-attempt-to-reduce-variance-by-increasing-bias." id="toc-some-estimators-attempt-to-reduce-variance-by-increasing-bias."><span class="toc-section-number">6</span> Some estimators attempt to reduce variance by increasing bias.</a>
+<li><a href="#some-estimators-attempt-to-reduce-variance-by-increasing-bias." id="toc-some-estimators-attempt-to-reduce-variance-by-increasing-bias."><span class="toc-section-number">6</span> 6. Some estimators attempt to reduce variance by increasing bias.</a>
 <ul>
 <li><a href="#design-based-estimators" id="toc-design-based-estimators"><span class="toc-section-number">6.1</span> Design-based estimators</a></li>
 <li><a href="#linear-regression-estimators" id="toc-linear-regression-estimators"><span class="toc-section-number">6.2</span> Linear regression estimators</a></li>
 <li><a href="#multilevel-model-estimators" id="toc-multilevel-model-estimators"><span class="toc-section-number">6.3</span> Multilevel model estimators</a></li>
 </ul></li>
-<li><a href="#for-each-estimator-that-achieves-a-point-estimator-there-may-be-multiple-options-for-estimating-standard-errors." id="toc-for-each-estimator-that-achieves-a-point-estimator-there-may-be-multiple-options-for-estimating-standard-errors."><span class="toc-section-number">7</span> For each estimator that achieves a point estimator, there may be multiple options for estimating standard errors.</a></li>
-<li><a href="#the-analysts-choices-of-estimand-estimator-and-standard-error-estimator-matter-in-some-cases-and-matter-less-in-others." id="toc-the-analysts-choices-of-estimand-estimator-and-standard-error-estimator-matter-in-some-cases-and-matter-less-in-others."><span class="toc-section-number">8</span> The analyst’s choices of estimand, estimator, and standard error estimator matter in some cases, and matter less in others.</a>
+<li><a href="#for-each-estimator-that-achieves-a-point-estimator-there-may-be-multiple-options-for-estimating-standard-errors." id="toc-for-each-estimator-that-achieves-a-point-estimator-there-may-be-multiple-options-for-estimating-standard-errors."><span class="toc-section-number">7</span> 7. For each estimator that achieves a point estimator, there may be multiple options for estimating standard errors.</a></li>
+<li><a href="#the-analysts-choices-of-estimand-estimator-and-standard-error-estimator-matter-in-some-cases-and-matter-less-in-others." id="toc-the-analysts-choices-of-estimand-estimator-and-standard-error-estimator-matter-in-some-cases-and-matter-less-in-others."><span class="toc-section-number">8</span> 8. The analyst’s choices of estimand, estimator, and standard error estimator matter in some cases, and matter less in others.</a>
 <ul>
 <li><a href="#point-estimates" id="toc-point-estimates"><span class="toc-section-number">8.1</span> Point estimates</a></li>
 <li><a href="#standard-errors" id="toc-standard-errors"><span class="toc-section-number">8.2</span> Standard errors</a></li>
 </ul></li>
-<li><a href="#the-choice-of-estimator-impacts-power." id="toc-the-choice-of-estimator-impacts-power."><span class="toc-section-number">9</span> The choice of estimator impacts power.</a></li>
-<li><a href="#takeaway-advice-for-researchers-on-multisite-trials" id="toc-takeaway-advice-for-researchers-on-multisite-trials"><span class="toc-section-number">10</span> Takeaway advice for researchers on multisite trials</a></li>
+<li><a href="#the-choice-of-estimator-impacts-power." id="toc-the-choice-of-estimator-impacts-power."><span class="toc-section-number">9</span> 9. The choice of estimator impacts power.</a></li>
+<li><a href="#takeaway-advice-for-researchers-on-multisite-trials" id="toc-takeaway-advice-for-researchers-on-multisite-trials"><span class="toc-section-number">10</span> 10. Takeaway advice for researchers on multisite trials</a></li>
 <li><a href="#references" id="toc-references">References</a></li>
 </ul>
 </div>
 
 <!-- title: "10 Things to Know About Multisite or Block-Randomized Trials" -->
 <!-- author: "Methods Guide Author: Kristen Hunter" -->
-<div id="introduction" class="section level1" number="1">
-<h1><span class="header-section-number">1</span> Introduction</h1>
-<div id="what-is-a-multisite-or-block-randomized-trial" class="section level2" number="1.1">
-<h2><span class="header-section-number">1.1</span> What is a multisite or block randomized trial?</h2>
+<div id="what-is-a-multisite-or-block-randomized-trial" class="section level1" number="1">
+<h1><span class="header-section-number">1</span> 1. What is a multisite or block randomized trial?</h1>
 <p>A multisite or block-randomized trial is a randomized experiment “in which sample members are randomly assigned to a program or a control group <em>within</em> each of a number of sites” (<span class="citation">S. W. Raudenbush and Bloom (<a href="#ref-Raudenbush2015" role="doc-biblioref">2015</a>)</span>).</p>
-<p>For illustration, this guide will focus on multisite educational trials, although multisite trials are not unique to education.
+<p>This guide focuses on multisite educational trials for illustration, although multisite trials are not unique to education.
 Multisite trials are a subset of multilevel randomized controlled trials (RCTs), in which units are nested within hierarchical structures, such as students nested within schools nested within districts.
 This guide uses as an illustrative example the case where each site is a school, although they could also be districts or classrooms; thus the term “site” and “school” are used interchangeably.</p>
 <p>An advantage of multisite trials is that they allow a researcher to study average impact across
 units or sites, while also getting a sense of heterogeneity across sites (<span class="citation">S. W. Raudenbush and Bloom (<a href="#ref-Raudenbush2015" role="doc-biblioref">2015</a>)</span>).
 However, the opportunities provided by multisite trials also come with their own challenges.
 Much of the rest of this guide will discuss the choices that researchers must make when analyzing multisite trials, and the consequences of these choices.</p>
-</div>
-<div id="preliminaries-estimands-estimators-and-estimates" class="section level2" number="1.2">
-<h2><span class="header-section-number">1.2</span> Preliminaries: estimands, estimators, and estimates</h2>
+<div id="preliminaries-estimands-estimators-and-estimates" class="section level2" number="1.1">
+<h2><span class="header-section-number">1.1</span> Preliminaries: estimands, estimators, and estimates</h2>
 <p>Before diving in, let’s introduce the definitions of estimand, estimator, and estimate.
 These concepts are sometimes conflated, but disentangling them increases clarity and understanding.
 The main distinction is that the <em>estimand</em> is the goal, while the <em>estimator</em> is the analysis we do in order to reach that goal.</p>
@@ -546,28 +549,28 @@ <h2><span class="header-section-number">1.2</span> Preliminaries: estimands, est
 <p>Given an estimand, the researchers choose their <em>estimator</em> to be the coefficient from an OLS regression of the observed outcome on site-specific fixed effects and the treatment indicator.
 To calculate standard errors, they use Huber-White robust standard errors.
 All these choices result in a point <em>estimate</em> (e.g. the program increased reading scores by <span class="math inline">\(5\)</span> points) and a measure of uncertainty (e.g. a standard error of <span class="math inline">\(2\)</span> points).</p>
-<p>Next, we’ll also need some notation.
+<p>We’ll also need some notation.
 This guide follows the Neyman-Rubin potential outcomes notation (<span class="citation">Splawa-Neyman, Dabrowska, and Speed (<a href="#ref-Neyman1923" role="doc-biblioref">1923/1990</a>)</span>, <span class="citation">Imbens and Rubin (<a href="#ref-Imbens2015" role="doc-biblioref">2015</a>)</span>).
-The outcomes are <span class="math inline">\(Y_{ij}\)</span> for unit <span class="math inline">\(i\)</span> in site <span class="math inline">\(j\)</span>.
-The potential outcomes are <span class="math inline">\(Y_{ij}(1)\)</span>: the outcome given active treatment, and <span class="math inline">\(Y_{ij}(0)\)</span>: the outcome given control treatment.
+The observed outcomes are <span class="math inline">\(Y_{ij}\)</span> for unit <span class="math inline">\(i\)</span> in site <span class="math inline">\(j\)</span>.
+The potential outcomes are <span class="math inline">\(Y_{ij}(1)\)</span>, the outcome given active treatment, and <span class="math inline">\(Y_{ij}(0)\)</span>, the outcome given control treatment.
 The quantity <span class="math inline">\(B_{ij}\)</span> is the unit-level intention-to-treat effect (ITT) <span class="math inline">\(B_{ij} = Y_{ij}(1) - Y_{ij}(0)\)</span>.
 If there is no noncompliance, the ITT is the ATE, as defined above.
 Then <span class="math inline">\(B_j\)</span> is the average impact at site <span class="math inline">\(j\)</span>, <span class="math inline">\(B_j = 1/N_j \sum_{i = 1}^{N_j} B_{ij}\)</span> where <span class="math inline">\(N_j\)</span> is the number of units at site <span class="math inline">\(j\)</span>.
 Finally, <span class="math inline">\(N = \sum_{j = 1}^{J} N_j\)</span>.</p>
 <p>This guide is structured around the choices an analyst must make concerning estimand and estimators, and the resulting consequences.
 The choice of estimand impacts the substantive conclusion that a researcher makes.
-The choice of estimator and standard error estimator results in different statistical properties, including a potential tradeoff between bias and variance.
+The choice of estimator and standard error estimator results in different statistical properties, including a potential trade off between bias and variance.
 This guide summarizes material using the framework provided by <span class="citation">Miratrix, Weiss, and Henderson (<a href="#ref-Miratrix2020" role="doc-biblioref">2021</a>)</span>.</p>
 </div>
 </div>
 <div id="a-multisite-trial-is-a-type-of-a-blocked-or-stratified-randomized-experiment." class="section level1" number="2">
-<h1><span class="header-section-number">2</span> A multisite trial is a type of a blocked or stratified randomized experiment.</h1>
+<h1><span class="header-section-number">2</span> 2. A multisite trial is a type of a blocked or stratified randomized experiment.</h1>
 <div id="a-multisite-trial-is-fundamentally-a-blocked-or-stratified-rct." class="section level2" number="2.1">
 <h2><span class="header-section-number">2.1</span> A multisite trial is fundamentally a blocked or stratified RCT.</h2>
 <p>A multisite trial is a blocked RCT with 2 levels: randomization occurs at the student level (level 1) within blocks defined by sites/schools (level 2).
 For example, in a study of a new online math tool for high school students, randomization occurs at the student level within blocks defined by sites/schools.
 Perhaps half of students at each school are assigned to the status quo / control treatment (no additional math practice),
-and half are assigned to theactive treatment (an offer of additional math practice at home using an online tool).</p>
+and half are assigned to the active treatment (an offer of additional math practice at home using an online tool).</p>
 <p>Because of the direct correspondence between multisite trials and blocked experiments, statistical properties of blocked experiments also translate directly to multisite experiments.
 The main difference between a traditional blocked RCT and a multisite experiment is that in many blocked RCTs, the researcher is able to choose the blocks.
 For example, in a clinical trial, a researcher may decide to block based on gender or specific age categories.
@@ -576,23 +579,22 @@ <h2><span class="header-section-number">2.1</span> A multisite trial is fundamen
 Multisite experiments have structural blocks, such as districts, schools, or classrooms.
 The type of block can impact variance estimation, as shown in <span class="citation">Pashley and Miratrix (<a href="#ref-Pashley2021" role="doc-biblioref">2021</a>)</span> and <span class="citation">Pashley and Miratrix (<a href="#ref-Pashley2022" role="doc-biblioref">2022</a>)</span>.</p>
 <p>The <a href="https://egap.org/our-work-0/the-metaketa-initiative/" target="_blank">EGAP Metaketa Projects</a>
-are also multisite trials: the 5–7 countries that contain sites for each study are fixed and chosen in advance by the different research teams.</p>
+are also multisite trials: the 5 to 7 countries that contain sites for each study are fixed and chosen in advance by the different research teams.</p>
 </div>
 <div id="a-multisite-trial-is-not-a-cluster-randomized-trial" class="section level2" number="2.2">
 <h2><span class="header-section-number">2.2</span> A multisite trial is not a cluster-randomized trial</h2>
 <p>A different type of RCT is a <a href="https://egap.org/resource/10-things-to-know-about-cluster-randomization/" target="_blank">cluster-randomized
-design</a>
-,
+design</a>,
 in which entire schools are assigned to either the active treatment or control
 treatment.
 <a href="https://youtu.be/bL2U9z8hX1k" target="_blank">This video explains the difference between cluster and
 block-randomized designs</a>.
-In a multisite trial trial, treatment is assigned <strong>within a block to individual units</strong>.
+In a multisite trial, treatment is assigned <strong>within a block to individual units</strong>.
 In a cluster-randomized trial, treatment is assigned to <strong>groups</strong> of units.
 Some designs <a href="https://declaredesign.org/r/designlibrary/reference/block_cluster_two_arm_designer.html" target="_blank">combine cluster- and
-block-randomization</a>
-.</p>
-<p>Another design that is not a multisite or block-randomized trial is an experiment that takes place in only one school and assigns individual students to active treatment and control treatment: this study has only one site and thus differences
+block-randomization</a>.</p>
+<p>Another design that is not a multisite or block-randomized trial is an experiment that takes place in only one school and assigns individual students to active treatment and control treatment.
+This type of study has only one site and thus differences
 between sites do not matter in this design.</p>
 </div>
 <div id="why-choose-a-multisite-or-block-randomized-trial-design" class="section level2" number="2.3">
@@ -600,9 +602,9 @@ <h2><span class="header-section-number">2.3</span> Why choose a multisite or blo
 <p>In most contexts, blocking reduces estimation error over an unblocked (completely randomized) experiment <span class="citation">(<a href="#ref-moore2012multivariate" role="doc-biblioref">Moore 2012</a>; <a href="#ref-gerber2012field" role="doc-biblioref">Gerber and Green 2012</a>)</span>.
 Thus, blocked experiments generally offer higher statistical power than unblocked experiments.
 Blocking is most helpful in increasing precision and statistical power in the setting where there is variation in the outcome, and where the blocks are related to this variation.</p>
-<p>In multisite trials as compared to block-randomized trials, the researcher cannot purposely construct blocks to reduce variation, because they are defined by pre-existing sites.
+<p>In multisite trials as compared to block-randomized trials, the researcher typically cannot purposely construct blocks to reduce variation, because they are defined by pre-existing sites.
 However, the researcher can hope, and often expect, that sites naturally explain some between-site variation.
-For example, if some schools tend to have larger impacts than others, and the size of the impact is related to the average income of families attending that school, then blocked randomization using the school as a block improves efficiency over complete randomization.</p>
+For example, if some schools tend to have higher outcomes than others, then blocked randomization using the school as a block improves efficiency over complete randomization.</p>
 <p>Randomizing with purposefully created blocks or pre-existing sites also helps analysts learn about how treatment effects may vary across the sites or groups of people categorized into the blocks.
 If a new treatment should help the lowest performing students most, but in any given study most students are not the lowest performing, then researchers may prefer to create blocks of students within schools with the students divided by their previous performance.
 This blocking within site would allow comparisons of the treatment effects on the relatively rare lowest performing students with the treatment effects on the relatively rare highest performing students.</p>
@@ -613,11 +615,11 @@ <h2><span class="header-section-number">2.4</span> Why not block?</h2>
 In other studies, the construction and choice of blocking criteria is a choice.
 <span class="citation">Pashley and Miratrix (<a href="#ref-Pashley2022" role="doc-biblioref">2022</a>)</span> shows that blocking is generally beneficial, but also explores settings in which it may be harmful.
 Blocking does result in fewer degrees of freedom, but in practice this reduction is rarely an issue, unless an experiment is very small <span class="citation">(<a href="#ref-Imai2008" role="doc-biblioref">Imai, King, and Stuart 2008</a>)</span>.
-Any use of blocking requires that an analyst keep track of the blocks and also that an analyst reflect the blocks in subsequent analysis: in many circumstances estimating average treatment effects from a block-randomized experiment while ignoring the blocks will yield biased estimates of the underlying targeted estimands (See <a href="https://declaredesign.org/blog/biased-fixed-effects.html">“The trouble with ‘controlling for blocks’”</a> and <a href="https://egap.org/resource/sd-block-rand">“Estimating Average Treatment Effects in Block Randomized Experiments”</a> for demostrations of bias arising from different approaches to weighting by blocks.)</p>
+Any use of blocking requires that an analyst keep track of the blocks and also that an analyst reflect the blocks in subsequent analysis: in many circumstances estimating average treatment effects from a block-randomized experiment while ignoring the blocks will yield biased estimates of the underlying targeted estimands (see <a href="https://declaredesign.org/blog/biased-fixed-effects.html">“The trouble with ‘controlling for blocks’”</a> and <a href="https://egap.org/resource/sd-block-rand">“Estimating Average Treatment Effects in Block Randomized Experiments”</a> for demonstrations of bias arising from different approaches to weighting by blocks).</p>
 </div>
 </div>
 <div id="analysis-can-either-target-the-population-in-the-experiment-or-a-broader-population." class="section level1" number="3">
-<h1><span class="header-section-number">3</span> Analysis can either target the population in the experiment, or a broader population.</h1>
+<h1><span class="header-section-number">3</span> 3. Analysis can either target the population in the experiment, or a broader population.</h1>
 <p>The first choice a researcher must make in defining their estimand is the population of interest.
 The researcher may want to focus on the <strong>finite population</strong>: only those units in the experimental pool or sample.
 Alternatively, they can expand their estimand to consider the <strong>super population</strong>.
@@ -645,7 +647,7 @@ <h1><span class="header-section-number">3</span> Analysis can either target the
 <p>For more discussion of the consequence of the super population and finite population frameworks, see <span class="citation">Schochet (<a href="#ref-Schochet2016" role="doc-biblioref">2016</a>)</span> and <span class="citation">Pashley and Miratrix (<a href="#ref-Pashley2021" role="doc-biblioref">2021</a>)</span>.</p>
 </div>
 <div id="the-average-site-effect-is-not-the-same-as-the-average-person-effect." class="section level1" number="4">
-<h1><span class="header-section-number">4</span> The average site effect is not the same as the average person effect.</h1>
+<h1><span class="header-section-number">4</span> 4. The average site effect is not the same as the average person effect.</h1>
 <p>The second choice a researcher makes is the target of inference: is the researcher interested in the <strong>average student</strong>, or
 the <strong>average site</strong> (<span class="citation">Miratrix, Weiss, and Henderson (<a href="#ref-Miratrix2020" role="doc-biblioref">2021</a>)</span>)?</p>
 <p>When we consider the average student impact, we weigh each student equally.
@@ -665,11 +667,11 @@ <h1><span class="header-section-number">4</span> The average site effect is not
 \frac{1}{J} \sum_{j = 1}^{J} B_j.
 \]</span>
 Note that in the case where all sites are of the same size, or all sites have the same impact, then these two estimands are the same.</p>
-<p>To summarize the previous two sections, there have been two axes of choices: the population of interest (FP or SP for finite and super population), and the target of inference (persons or sites).
+<p>To summarize, this section and the prior section have given two axes of choices: the population of interest (FP or SP for finite and super population), and the target of inference (persons or sites).
 These choices result in four possible estimands: FP-persons, SP-persons, FP-sites, and SP-sites.</p>
 </div>
 <div id="there-are-many-widely-used-estimators-that-target-the-same-estimands-including-design-based-linear-regression-and-multilevel-models." class="section level1" number="5">
-<h1><span class="header-section-number">5</span> There are many widely-used estimators that target the same estimands, including design-based, linear regression, and multilevel models.</h1>
+<h1><span class="header-section-number">5</span> 5. There are many widely-used estimators that target the same estimands, including design-based, linear regression, and multilevel models.</h1>
 <p>After choosing an <em>estimand</em>, the researcher must then choose an <em>estimator</em>, a process to arrive at the estimate of interest.
 There are three main categories of estimators: <strong>design based</strong>, <strong>linear regression</strong>, and <strong>multilevel modeling</strong>.
 Linear regression and multilevel modeling are both model-based approaches to statistical inference.<a href="#fn1" class="footnote-ref" id="fnref1"><sup>1</sup></a>
@@ -678,9 +680,10 @@ <h1><span class="header-section-number">5</span> There are many widely-used esti
 <p>The different categories of estimator differ both philosophically and practically.
 Each category assumes a different source of randomness, and thus has a different statistical justification.</p>
 <p><strong>Design-based</strong> estimators specifically target the four estimands outlined above.
-The only source of uncertainty is assumed to be the treatment assignment: which units happened to be assigned to the active treatment, and which happened to be assigned to the control treatment.
+The main source of uncertainty is assumed to be the treatment assignment: which units happened to be assigned to the active treatment, and which happened to be assigned to the control treatment.
 This assumption is the reason for their name; the uncertainty in the estimates is by design, from the purposeful randomization of units.
-Using design-based estimators is also sometimes called Neymanian inference, as the estimators and properties were first introduced by Neyman (<span class="citation">Splawa-Neyman, Dabrowska, and Speed (<a href="#ref-Neyman1923" role="doc-biblioref">1923/1990</a>)</span>).</p>
+Using design-based estimators is also sometimes called Neymanian inference, as the estimators and properties were first introduced by Neyman (<span class="citation">Splawa-Neyman, Dabrowska, and Speed (<a href="#ref-Neyman1923" role="doc-biblioref">1923/1990</a>)</span>).
+Design-based estimators can also incorporate uncertainty from sampling when using a super population framework.</p>
 <p><strong>Linear regression</strong> estimators are the most familiar to many researchers.
 With these estimators, the observed outcomes are assumed to be a linear function of the treatment assignment, (optionally) site-specific effects, (optionally) covariates, and random error.
 In standard regression theory, the only source of randomness is the error term.
@@ -698,8 +701,8 @@ <h1><span class="header-section-number">5</span> There are many widely-used esti
 For a more comprehensive look at multilevel models, see <span class="citation">S. W. Raudenbush and Bloom (<a href="#ref-Raudenbush2015" role="doc-biblioref">2015</a>)</span>.</p>
 <p>Let’s examine a few popular models among linear regression and multilevel models in more detail.
 Note that these models as presented do not include covariates, but covariates can easily be incorporated to increase power if the analyst is willing to increase bias by a small amount in exchange (often a very small amount if the experiment is large enough) <span class="citation">(<a href="#ref-lin2013agnostic" role="doc-biblioref">Lin 2013</a>)</span>.</p>
-<div id="linear-regression-model-assumptions" class="section level2" number="5.1">
-<h2><span class="header-section-number">5.1</span> Linear regression model assumptions</h2>
+<div id="common-linear-regression-models" class="section level2" number="5.1">
+<h2><span class="header-section-number">5.1</span> Common linear regression models</h2>
 <p><strong>Fixed effects with a constant treatment (FE)</strong></p>
 <p>With this model, the researcher assumes that there are site-specific fixed effects (intercepts), but a common overall ATE.
 The assumed model is
@@ -716,19 +719,20 @@ <h2><span class="header-section-number">5.1</span> Linear regression model assum
 \]</span>
 Given a series of site-specific treatment estimates <span class="math inline">\(\hat{\beta}_j\)</span>, these estimates are then averaged, with weights by either simple weighting (see <span class="citation">Clark and Silverberg (<a href="#ref-Clark2011" role="doc-biblioref">2011</a>)</span>) or by site size.</p>
 </div>
-<div id="multilevel-model-assumptions" class="section level2" number="5.2">
-<h2><span class="header-section-number">5.2</span> Multilevel model assumptions</h2>
-<p>Once an analyst selects a multilevel model, for site intercepts and site impacts they must decide: what is considered random, and what is considered fixed?</p>
+<div id="common-multilevel-models" class="section level2" number="5.2">
+<h2><span class="header-section-number">5.2</span> Common multilevel models</h2>
+<p>Once an analyst selects multilevel modeling, for site intercepts and site impacts they must decide: what is considered random, and what is considered fixed?</p>
 <p><strong>Fixed intercept, random treatment coefficient (FIRC)</strong></p>
 <p>This model is similar to the fixed effects models above, but assumes that the site impact <span class="math inline">\(\beta_j\)</span> is drawn from a shared distribution.
-<span class="math display">\[\begin{align*}
+The FIRC model was more recently designed to handle bias issues that arise when the proportion of units treated varies across sites.</p>
+<p><span class="math display">\[\begin{align*}
 \text{Level 1}\qquad &amp; Y_{ij} = \sum_{k = 1}^{J} \alpha_k
 \text{Site}_{k,ij} + \beta_j T_{ij} + e_{ij}\\
 \text{Level 2}\qquad &amp; \beta_j = \beta + b_j
 \end{align*}\]</span>
 See <span class="citation">S. W. Raudenbush and Bloom (<a href="#ref-Raudenbush2015" role="doc-biblioref">2015</a>)</span> and <span class="citation">Bloom and Porter (<a href="#ref-Bloom2017" role="doc-biblioref">2017</a>)</span>.</p>
 <p><strong>Random intercept, random treatment coefficient (RIRC)</strong></p>
-<p>This model further generalizes to assume that both the site intercept and site impact are drawn from shared distributions.
+<p>This model is an older version of multilevel models, and assumes that both the site intercept and site impact are drawn from shared distributions.
 <span class="math display">\[\begin{align*}
 \text{Level 1}\qquad &amp; Y_{ij} = A_j + \beta_j T_{ij} + e_{ij}\\
 \text{Level 2}\qquad &amp; \beta_j = \beta + b_j\\
@@ -749,15 +753,14 @@ <h2><span class="header-section-number">5.2</span> Multilevel model assumptions<
 </div>
 </div>
 <div id="some-estimators-attempt-to-reduce-variance-by-increasing-bias." class="section level1" number="6">
-<h1><span class="header-section-number">6</span> Some estimators attempt to reduce variance by increasing bias.</h1>
+<h1><span class="header-section-number">6</span> 6. Some estimators attempt to reduce variance by increasing bias.</h1>
 <p>Each category of estimator (design, regression, and multilevel) results in a different estimation approach.
 One way to characterize the categories is the weights induced by the choice of estimator.
 The properties of each estimator also result in different consequences for bias and variance.
-Design-based estimators are unbiased, but may not always afford the most precise estimates.
+Design-based estimators are generally unbiased, but may not always afford the most precise estimates.
 In general, model-based estimators trade bias for variance.
 Thus, they can sometimes have a lower mean squared error than design-based estimators.
 One way that model-based estimators increase precision is through the easy incorporation of covariates.
-Although design-based estimators can also incorporate covariates, it is not always as straightforward.
 Covariate adjustment methods that incorporate covariates result in the equivalent to a weighted regression approach.</p>
 <div id="design-based-estimators" class="section level2" number="6.1">
 <h2><span class="header-section-number">6.1</span> Design-based estimators</h2>
@@ -767,10 +770,10 @@ <h2><span class="header-section-number">6.1</span> Design-based estimators</h2>
 <p>The design-based estimators are
 <span class="math display">\[\begin{align*}
 \hat{\beta}_{DB-persons} &amp;= \sum_{j = 1}^{J} \frac{N_j}{N} \hat{B_j} \\
-\hat{\beta}_{DB-sites} &amp;= \sum_{j = 1}^{J} \frac{1}{J} \hat{B_j}
+\hat{\beta}_{DB-sites} &amp;= \sum_{j = 1}^{J} \frac{1}{J} \hat{B_j}.
 \end{align*}\]</span>
 Design-based estimators are generally <em>unbiased</em> for their corresponding estimands (person-weighted or site-weighted).
-Unbiasedness does not hold for one superpopulation model; see <span class="citation">Pashley and Miratrix (<a href="#ref-Pashley2022" role="doc-biblioref">2022</a>)</span> for more details.</p>
+Unbiasedness does not hold for one super population model; see <span class="citation">Pashley and Miratrix (<a href="#ref-Pashley2022" role="doc-biblioref">2022</a>)</span> for more details.</p>
 </div>
 <div id="linear-regression-estimators" class="section level2" number="6.2">
 <h2><span class="header-section-number">6.2</span> Linear regression estimators</h2>
@@ -781,13 +784,12 @@ <h2><span class="header-section-number">6.2</span> Linear regression estimators<
 \hat{\beta}_{FE} = \sum_{j = 1}^{J} \frac{N_j p_j (1 - p_j)}{Z} \hat{B_j},
 \]</span>
 where <span class="math inline">\(p_j\)</span> is the proportion treated at site <span class="math inline">\(j\)</span>.
-The quantity <span class="math inline">\(Z\)</span> is a normalizing constant: <span class="math inline">\(Z = \sum_{j = 1}^{J} N_j p_j (1-p_j)\)</span> to ensure the weights sum to one.
-In this model, the weights include <span class="math inline">\(p_j\)</span>, which tells us information about the precision of the estimate for that site:
-<span class="math inline">\(N_j p_j (1 - p_j)\)</span> is the inverse of <span class="math inline">\(Var(\hat{\beta_j})\)</span>.
-This expression shows that sites with larger <span class="math inline">\(N_j\)</span>, or have <span class="math inline">\(p_j\)</span> closer to <span class="math inline">\(0.5\)</span>, have larger weights.</p>
+The quantity <span class="math inline">\(Z\)</span> is a normalizing constant, so <span class="math inline">\(Z\)</span> is defined as <span class="math inline">\(\sum_{j = 1}^{J} N_j p_j (1-p_j)\)</span> to ensure the weights sum to one.
+The weights are <span class="math inline">\(N_j p_j (1 - p_j)\)</span>, which is the inverse of <span class="math inline">\(Var(\hat{\beta_j})\)</span>, so the weights are related to the precision of the estimate for each site.
+This expression shows that sites with larger <span class="math inline">\(N_j\)</span>, or that have <span class="math inline">\(p_j\)</span> closer to <span class="math inline">\(0.5\)</span>, have larger weights.</p>
 <p>The FE estimator is not generally unbiased for either person-weighted or site-weighted estimands.
 If the impact size <span class="math inline">\(B_j\)</span> is related to the weights (<span class="math inline">\(N_j p_j (1 - p_j)\)</span>), then the estimator could be biased.
-For example, if sites that treat a higher proportion of treated units also have a large impact, then <span class="math inline">\(B_j\)</span> can be related to <span class="math inline">\(p_j (1- p_j)\)</span>.
+For example, if sites that treat a higher proportion of treated units also experience a larger treatment impact, then <span class="math inline">\(B_j\)</span> can be related to <span class="math inline">\(p_j (1- p_j)\)</span>.
 This setting is plausible for example if sites with more resources to intervene on more students also implement the intervention more effectively.
 If larger sites are more effective, then <span class="math inline">\(B_j\)</span> can be related to <span class="math inline">\(N_j p_j (1- p_j)\)</span>.</p>
 <p>Instead, the FE estimator is unbiased for an estimand that weights the site impacts by <span class="math inline">\(N_j p_j (1- p_j)\)</span>.
@@ -798,18 +800,20 @@ <h2><span class="header-section-number">6.2</span> Linear regression estimators<
 <div id="multilevel-model-estimators" class="section level2" number="6.3">
 <h2><span class="header-section-number">6.3</span> Multilevel model estimators</h2>
 <p>Multilevel models also result in precision weighting, but in these models the estimated precision also takes into account the assumed underlying variance in site impacts.
-For example, consider the FIRC model:
+For example, the FIRC model can be expressed roughly as:
 <span class="math display">\[
 \hat{\beta}_{ML-FIRC*} = \sum_{j = 1}^{J} \frac{1}{Z}
-\left(\frac{\sigma^2}{N_j p_j ( 1 - p_j)} + \tau^2\right)^{-1}
+\left(\frac{\sigma^2}{N_j p_j ( 1 - p_j)} + \tau^2\right)^{-1} \hat{B_j},
 \]</span>
 where <span class="math inline">\(Z\)</span> is again a normalizing constant, <span class="math inline">\(Z = \sum_{j = 1}^{J} \left(\frac{\sigma^2}{N_j p_j ( 1 - p_j)} + \tau^2\right)^{-1}\)</span>.
-This equation assumes that <span class="math inline">\(b_j\)</span> has known variance <span class="math inline">\(\tau^2\)</span>, and <span class="math inline">\(e_{ij}\)</span> has known variance <span class="math inline">\(\sigma^2\)</span>.
+This equation assumes that the <span class="math inline">\(b_j\)</span> have known variance <span class="math inline">\(\tau^2\)</span>, and the <span class="math inline">\(e_{ij}\)</span> have known variance <span class="math inline">\(\sigma^2\)</span>.
 In general, we do not know these quantities, and instead must estimate them.
 However, we can see that the implied precision weights incorporate the additional uncertainty assumed in the value of <span class="math inline">\(b_j\)</span>.</p>
 <p>The RIRC model imposes the same structure on the site impacts, and thus the weights are similar to the FIRC model.
-The RICC model assumes a constant treatment impact, and thus is essentially equivalent to the fixed effects with constant treatment model (FE) when it comes to estimating the site impacts.</p>
-<p>We summarize the weights below.</p>
+The RICC model assumes a constant treatment impact, and thus is essentially equivalent to the precision-weighted fixed effects with constant treatment model (FE) when it comes to estimating the site impacts.</p>
+<p>We summarize the weights in the table below.
+The following table includes additional estimators that are not discussed in this guide;
+for more information about these additional estimators, see <span class="citation">Miratrix, Weiss, and Henderson (<a href="#ref-Miratrix2020" role="doc-biblioref">2021</a>)</span>.</p>
 <table>
 <colgroup>
 <col width="33%" />
@@ -849,15 +853,15 @@ <h2><span class="header-section-number">6.3</span> Multilevel model estimators</
 </div>
 </div>
 <div id="for-each-estimator-that-achieves-a-point-estimator-there-may-be-multiple-options-for-estimating-standard-errors." class="section level1" number="7">
-<h1><span class="header-section-number">7</span> For each estimator that achieves a point estimator, there may be multiple options for estimating standard errors.</h1>
+<h1><span class="header-section-number">7</span> 7. For each estimator that achieves a point estimator, there may be multiple options for estimating standard errors.</h1>
 <p>The difference between the finite population and super population framework comes into focus when calculating the standard error of various estimators.
 In general, the super population framework results in larger estimates of error because of the additional uncertainty induced by assuming the sites observed are randomly drawn from a larger population.
-In general, variation can be characterized by either <em>within site</em> variation or <em>between site</em> variation.
+In general, variation in the control outcome can be broken down into <em>within site</em> variation and <em>between site</em> variation.
 In the finite population framework, estimators calculate variation <em>within</em> sites, and then estimators average this variation across sites.
 In the super population framework, estimators look at the variation <em>between</em> sites to “capture both any within-site estimation error along with the uncertainty associated with sampling sites from a larger population” (<span class="citation">Miratrix, Weiss, and Henderson (<a href="#ref-Miratrix2020" role="doc-biblioref">2021</a>)</span>).
 For both approaches, modeling assumptions can stabilize uncertainty estimation procedures, but also risk inducing bias if the modeling assumptions are wrong.</p>
 <p>For design-based estimators, for the finite population framework Neyman developed a conservative estimator for the standard error using the observed outcomes.
-First, within-site uncertainty is estimated, and then these estimates are averaged with weights according to the target estimand.
+First, within-site uncertainty is estimated for each site, and then these estimates are averaged with weights according to the target estimand.
 The super population framework induces more complicated expressions that take into account the additional population variance.
 The details of standard errors for super population design-based estimators are beyond the scope of this guide.</p>
 <p>For linear regression estimators, the traditional way to calculate standard errors is using classical regression theory.
@@ -866,7 +870,7 @@ <h1><span class="header-section-number">7</span> For each estimator that achieve
 assumption (see <span class="citation">Weiss and Gupta (<a href="#ref-Weiss2019" role="doc-biblioref">2017</a>)</span> and <span class="citation">Richburg-Hayes and Bloom (<a href="#ref-Richburg-Hayes2008" role="doc-biblioref">2008</a>)</span>).
 Robust standard errors fall into a design-based approach instead of a model-based approach <span class="citation">(<a href="#ref-lin2013agnostic" role="doc-biblioref">Lin 2013</a>; Chapter 3 of <a href="#ref-gerber2012field" role="doc-biblioref">Gerber and Green 2012</a>)</span>.
 Huber-White standard errors correspond to the finite population framework, while the asymptotic theory justifying traditional cluster robust standard errors corresponds to the super population framework in regards the clusters. In a <a href="https://egap.org/resource/10-things-to-know-about-cluster-randomization/">cluster-randomized trial</a>, treatment is assigned to clusters, so there is also a finite-population-of-clusters perspective on cluster robust standard errors that is approximated in what are commonly known as CR2 standard errors <span class="citation">(<a href="#ref-Pustejovsky2018" role="doc-biblioref">Pustejovsky and Tipton 2018</a>)</span>.</p>
-<p>To briefly summarize this correspondence, first consider the motivation behind robust standard error estimators.
+<p>To briefly summarize the correspondence between standard error estimators and the assumed population, first consider the motivation behind robust standard error estimators.
 In the FE model, treatment effects are assumed to be constant across sites.
 Thus, if there is truly treatment effect heterogeneity, units in different sites will have different amounts of variation, and this variation will be incorporated into the error term.
 The assumption of <span class="math inline">\(iid\)</span> standard errors will be broken.
@@ -882,13 +886,13 @@ <h1><span class="header-section-number">7</span> For each estimator that achieve
 finite population framework because the treatment impacts are not assumed to be drawn from a super population, as they are assumed to be consistent across sites.</p>
 </div>
 <div id="the-analysts-choices-of-estimand-estimator-and-standard-error-estimator-matter-in-some-cases-and-matter-less-in-others." class="section level1" number="8">
-<h1><span class="header-section-number">8</span> The analyst’s choices of estimand, estimator, and standard error estimator matter in some cases, and matter less in others.</h1>
+<h1><span class="header-section-number">8</span> 8. The analyst’s choices of estimand, estimator, and standard error estimator matter in some cases, and matter less in others.</h1>
 <p>After discussing the different choices a researcher can make in analyzing a multisite trial, a big question remains: how do these choices impact empirical results?
 Which of these choices have a substantial impact on the conclusion we reach, and which do not matter as much?
-<span class="citation">Miratrix, Weiss, and Henderson (<a href="#ref-Miratrix2020" role="doc-biblioref">2021</a>)</span> conducted an empirical study to investigate these questions using 12 large multisite trials.</p>
+<span class="citation">Miratrix, Weiss, and Henderson (<a href="#ref-Miratrix2020" role="doc-biblioref">2021</a>)</span> conducted an empirical study to investigate these questions using 12 large multisite trials, backed up by simulation studies in certain cases.</p>
 <div id="point-estimates" class="section level2" number="8.1">
 <h2><span class="header-section-number">8.1</span> Point estimates</h2>
-<p>First, they consider the impact on point estimates.
+<p>First, they consider the impact of choices on point estimates.
 The authors ask, “to what extent can the choice of estimator of the overall average treatment effect result in a different impact estimate?”
 In general, the authors find that the choice of estimator can substantially impact the point estimates, although the degree of impact depends on the choice.
 The authors reach the following conclusions.</p>
@@ -900,29 +904,26 @@ <h2><span class="header-section-number">8.1</span> Point estimates</h2>
 <p><strong>For person-weighted estimands, the choice of estimator generally does not matter.</strong></p>
 <p>The unbiased design-based estimator and the precision-weighted fixed effect estimate both target the person-weighted estimand.
 There was little difference in estimates between these estimators.
-Most likely, “this implies that the potential bias in the bias-precision tradeoff to the fixed effect estimators is negligible in practice.”
-Other <a href="https://egap.org/resource/sd-block-rand/" target="_blank">authors</a> have been able to create situations in which the bias-precision tradeoff is more severe.</p>
+Most likely, “this implies that the potential bias in the bias-precision trade off to the fixed effect estimators is negligible in practice.”
+Other <a href="https://egap.org/resource/sd-block-rand/" target="_blank">authors</a> have been able to create situations in which the bias-precision trade off is more severe.</p>
 <p><strong>For site-weighted estimands, the choice of estimator can matter.</strong></p>
 <p>FIRC estimates did differ from the unbiased design-based site estimator.
-FIRC can be seen as an adaptive estimator: when there is little observed variation between sites, it tends to be more similar to person-weighted estimate instead of the site-weighted estimate.</p>
-<p><strong>Different estimators have different bias-variance tradeoffs.</strong></p>
-<p>Finally, the authors consider the empirical bias-variance tradeoff of different estimators, and find:</p>
+FIRC can be seen as an adaptive estimator: when there is little estimated variation in impacts between sites, it tends to be more similar to the person-weighted estimate instead of the site-weighted estimate.</p>
+<p><strong>Different estimators have different bias-variance trade offs.</strong></p>
+<p>Finally, the authors consider the empirical bias-variance trade off of different estimators, and find:</p>
 <ul>
 <li>FE estimators have little bias, but also do not improve precision much over design-based estimators.</li>
-<li>To further investigate, they conducted a simulation study and found:
-<ul>
 <li>FIRC tends to have lower mean squared error than design-based estimators.</li>
 <li>Larger site impact heterogeneity results in more biased estimates for FIRC.</li>
 <li>Even with more site impact heterogeneity, the mean squared error for FIRC estimators is still generally lower.</li>
 <li>Coverage for design-based estimators is more reliable, especially when site size is variable and site size is correlated with impact.</li>
-</ul></li>
 </ul>
 </div>
 <div id="standard-errors" class="section level2" number="8.2">
 <h2><span class="header-section-number">8.2</span> Standard errors</h2>
 <p>The second question concerns the choice of standard error estimators.
 The authors ask, “to what extent can the choice of estimator of the standard error of the overall average treatment effect result in a different estimated standard error?”</p>
-<p>Similar to for point estimates, the choice of standard error estimator can substantially impact the estimated standard error.
+<p>The choice of standard error estimator can substantially impact the estimated standard error.
 The authors reach the following conclusions.</p>
 <p><strong>The choice of estimand impacts the standard error.</strong></p>
 <p>Super population estimators generally have larger standard errors than finite population estimators.
@@ -939,30 +940,29 @@ <h2><span class="header-section-number">8.2</span> Standard errors</h2>
 </div>
 </div>
 <div id="the-choice-of-estimator-impacts-power." class="section level1" number="9">
-<h1><span class="header-section-number">9</span> The choice of estimator impacts power.</h1>
-<p>Given the discussion thus far, it is not surprising that modeling choices made by the analyst also impacts power.</p>
-<p>First, we define an important quantity in power calculations: the intraclass correlation coefficient (ICC).
-Broadly, variation can be categorized into <em>within</em>-site variation, and <em>between</em>-site variation.
-Blocking helps compensate for between-site variation.
-In educational trials, the ICC is the proportion of variance in the outcome that lies <em>between</em> sites (<span class="citation">Schochet (<a href="#ref-Schochet2016" role="doc-biblioref">2016</a>)</span>).
+<h1><span class="header-section-number">9</span> 9. The choice of estimator impacts power.</h1>
+<p>Given the discussion thus far, it is not surprising that modeling choices made by the analyst also impacts statistical power.</p>
+<p>To further understand power, we define an important quantity in power calculations: the intraclass correlation coefficient (ICC).
+Broadly, variation in the observed control outcomes can be categorized into <em>within</em>-site variation, and <em>between</em>-site variation.
+In educational trials, the ICC is the proportion of variation in the outcome that lies <em>between</em> sites (<span class="citation">Schochet (<a href="#ref-Schochet2016" role="doc-biblioref">2016</a>)</span>).
 The ICC is defined as the ratio of the variance at the site level divided by the overall variance of the individual outcomes.
 This quantity plays a different role in block-randomized trial power analysis depending on the target of inference chosen by the analyst.
 ICC is also used in the design and analysis of cluster-randomized trials.</p>
 <p>We consider two different estimators and how they impact power.
-First, consider a version of the FE model that has been expanded to include level 1 (student) covariates.
+First, consider a version of the finite population FE model that has been expanded to include level 1 (student) covariates.
 The standard error for the ATE estimator is</p>
 <p><span class="math display">\[
 SE = \sqrt{\frac{(1-\text{ICC})(1-R^2_{1})}{\bar{T}(1 - \bar{T}) J \bar{n}}},
 \]</span></p>
 <p>where <span class="math inline">\(ICC\)</span> is the intraclass correlation, <span class="math inline">\(R_1^2\)</span> is the proportion of variation explained by level 1 (student) covariates, <span class="math inline">\(\bar{T}\)</span> is the average number of treated units per site, <span class="math inline">\(J\)</span> is the number of sites, and <span class="math inline">\(\bar{n}\)</span> is the average number of students per site.
-or more information about this standard error expression, see the technical appendix of <span class="citation">Hunter, Miratrix, and Porter (<a href="#ref-Hunter2022" role="doc-biblioref">2022</a>)</span>.</p>
-<p>In contrast, consider the RIRC model.
+For more information about this standard error expression, see the technical appendix of <span class="citation">Hunter, Miratrix, and Porter (<a href="#ref-Hunter2022" role="doc-biblioref">2022</a>)</span>.</p>
+<p>In contrast, consider the super population RIRC model.
 The standard error for the ATE estimator is
 <span class="math display">\[
 SE = \sqrt{\frac{\text{ICC} \omega}{J} + \frac{(1-\text{ICC})(1-R^2_{1})}{\bar{T}(1 - \bar{T}) J \bar{n}}},
 \]</span>
-where <span class="math inline">\(\omega\)</span> is the ratio between the impact variation and the control outcome variation.
-We can see that in doing super population inference, the standard error has an additional term which is nonnegative, so it will be as least as large as the standard error from finite population inference.
+where <span class="math inline">\(\omega\)</span> is the ratio between the cross-site impact variation and the control outcome variation.
+We can see that in doing super population inference, the standard error has an additional term which is non-negative, so it will be as least as large as the standard error from finite population inference.
 A larger standard error will result in lower power.</p>
 <p>Examining these standard error formulae also gives a better understanding of what factors impact power.
 For example, having more explanatory power of student-level covariates (higher <span class="math inline">\(R_1^2\)</span>) decreases the standard error, and thus increases power.
@@ -974,7 +974,7 @@ <h1><span class="header-section-number">9</span> The choice of estimator impacts
 The newly-developed PUMP package (<span class="citation">Hunter, Miratrix, and Porter (<a href="#ref-Hunter2022" role="doc-biblioref">2022</a>)</span>) extends the functionality of PowerUpR! to experiments with multiple outcomes, in addition to providing user-friendly tools for exploring the sensitivity of power to different assumptions.</p>
 </div>
 <div id="takeaway-advice-for-researchers-on-multisite-trials" class="section level1" number="10">
-<h1><span class="header-section-number">10</span> Takeaway advice for researchers on multisite trials</h1>
+<h1><span class="header-section-number">10</span> 10. Takeaway advice for researchers on multisite trials</h1>
 <p>Many research plans and analyses do not clearly specify an estimand.
 This lack of clarity can both obscure the goal, and result in poor analysis choices.
 For example, different estimands imply different power analyses, but the choice is often not taken into account; super population estimands in particular result in larger standard errors, and thus often require larger sample sizes to be adequately powered.