Merge pull request #312 from pymc-labs/design-notation-explainer

Add experiment design notation docs page to new Knowledge Base section of the docs

Author: drbenvincent

docs/source/design_notation.md

@@ -0,0 +1,99 @@
+# Quasi-experimental design notation
+
+This page provides a concise summary of the tabular notation used by {cite:t}`shadish_cook_cambell_2002` and {cite:t}`reichardt2019quasi`. This notation provides a compact description of various experimental designs. While it is possible to describe randomised designs using this notation, we focus purely on {term}`quasi-experimental<Quasi-experiment>` designs here, with non-random allocation (abbreviated as `NR`). Observations are denoted by $O$. Time proceeds from left to right, so observations made through time are labelled as $O_1$, $O_2$, etc. The treatment is denoted by `X`. Rows represent different groups of units. Remember, a unit is a person, place, or thing that is the subject of the study.
+
+## Pretest-posttest designs
+
+One of the simplest designs is the pretest-posttest design. Here we have one row, denoting a single group of units. There is an `X` which means all are treated. The pretest is denoted by $O_1$ and the posttest by $O_2$. See p99 of {cite:t}`reichardt2019quasi`.
+
+|    |   |    |
+|----|---|----|
+$O_1$ | X | $O_2$ |
+
+Informally, if we think about drawing conclusions about the {term}`causal impact` of the treatment based on the change from $O_1$ to $O_2$, we might say that the treatment caused the change. However, this is a tenuous conclusion because we have no way of knowing what would have happened in the ({term}`counterfactual`) absence of the treatment.
+
+A variation of this design which may (slightly) improve this situation from the perspective of making causal claims, would be to take multiple pretest measures. This is shown below, see p107 of {cite:t}`reichardt2019quasi`.
+
+|    |  |   |    |
+|----|--|---|----|
+$O_1$ | $O_2$ | X | $O_3$ |
+
+This would allow us to estimate how the group was changing over time before the treatment was introduced. This could be used to make a stronger causal claim about the impact of the treatment. We could use {term}`interrupted time series<ITS>` analysis to help here.
+
+## Nonequivalent group designs
+
+In randomized experiments, with large enough groups, the randomization process should ensure that the treatment and control groups are approximately equivalent in terms of their attributes. This is positive for causal inference as we can be more sure that differences between control and test groups are due to treatment exposure, not because of differences in attributes of the groups.
+
+However, in quasi-experimental designs, with non-random (`NR`) allocation, we could expect there to be differences between the treatment and control groups' attributes. This poses some challenges in making strong causal claims about the impact of the treatment - we can't be sure that differences between the groups at the posttest are due to the treatment, or due to pre-existing differences between the groups.
+
+In the simplest {term}`nonequivalent group design<NEGD>`, we have two groups, one treated and one not treated, and just one posttest. See p114 of {cite:t}`reichardt2019quasi`.
+
+|     |   |    |
+|-----|---|----|
+| NR: | X | $O_1$ |
+| NR: |   | $O_1$ |
+
+The above design would be considered weak - the lack of a pre-test measure makes it hard to know whether differences between the groups at $O_1$ are due to the treatment or to pre-existing differences between the groups.
+
+This limitation can be addressed by adding a pretest measure. See p115 of {cite:t}`reichardt2019quasi`.
+
+|     |    |   |    |
+|-----|----|---|----|
+| NR: | $O_1$ | X | $O_2$ |
+| NR: | $O_1$ |   | $O_2$ |
+
+Non-equivalent group designs like this, with a pretest and a posttest measure could be analysed in a number of ways:
+1. **{term}`ANCOVA`:** Here, the group would be a categorical predictor (e.g. treated/untreated), the pretest measure would be a covariate (though there could be more than one), and the posttest measure would be the outcome.
+2. **{term}`Difference in differences`:** We can apply linear modeling approaches such as `y ~ group + time + group:time` to estimate the treatment effect. Here, `y` is the outcome measure, `group` is a binary variable indicating treatment or control group, and `time` is a binary variable indicating pretest or posttest. Note that this approach has a strong assumption of [parallel trends](https://en.wikipedia.org/wiki/Difference_in_differences#Assumptions) - that the treatment and control groups would have changed in the same way in the absence of the treatment.
+
+A limitation of the nonequivalent group designs with single pre and posttest measures is that we don't know how the groups were changing over time before the treatment was introduced. This can be addressed by adding multiple pretest measures and can help in assessing if the parallel trends assumption is reasonable. See p154 of {cite:t}`reichardt2019quasi`.
+
+|     |    |   | |    |
+|-----|----|---|-|----|
+| NR: | $O_1$ | $O_2$ | X | $O_3$ |
+| NR: | $O_1$ | $O_2$ |   | $O_3$ |
+
+Again, this design could be analysed using the difference-in-differences approach.
+
+## Interrupted time series designs
+
+While there is no control group, the {term}`interrupted time series design` is a powerful quasi-experimental design that can be used to estimate the causal impact of a treatment. The design involves multiple pretest and posttest measures. The treatment is introduced at a specific point in time, denoted by `X`. The design can be used to estimate the causal impact of the treatment by comparing the trajectory of the outcome variable before and after the treatment. See p203 of {cite:t}`reichardt2019quasi`.
+
+|     |    |   |    |   |    |    |    |    |
+|-----|----|---|----|---|----|----|----|----|
+| $O_1$ | $O_2$ | $O_3$ | $O_4$ | X | $O_5$ | $O_6$ | $O_7$ | $O_8$ |
+
+You can see that this is an example of a pretest-posttest design with multiple pre and posttest measures.
+
+## Comparative interrupted time series designs
+
+The {term}`comparative interrupted time-series<CITS>` design incorporates aspects of **interrupted time series** (with only a treatment group), and **nonequivalent group designs** (with a treatment and control group). This design can be used to estimate the causal impact of a treatment by comparing the trajectory of the outcome variable before and after the treatment in the treatment group, and comparing this to the trajectory of the outcome variable in the control group. See p226 of {cite:t}`reichardt2019quasi`.
+
+|     |    |   |    |   |    |    |    |    | |
+|-----|----|---|----|---|----|----|----|----|-|
+| NR: | $O_1$ | $O_2$ | $O_3$ | $O_4$ | X | $O_5$ | $O_6$ | $O_7$ | $O_8$ |
+| NR: | $O_1$ | $O_2$ | $O_3$ | $O_4$ |   | $O_5$ | $O_6$ | $O_7$ | $O_8$ |
+
+
+Because this design is very similar to the nonequivalent group design, simply with multiple pre and posttest measures, it is well-suited to analysis under the difference-in-differences approach.
+
+However, if we have many untreated units and one treated unit, then this design could be analysed with the {term}`synthetic control` approach.
+
+## Regression discontinuity designs
+
+The design notation for {term}`regression discontinuity designs<RDD>` are different from the others and take a bit of getting used to. We have two groups, but allocation to the groups are determined by a units' relation to a cutoff point `C` along a {term}`running variable`. Also, $O_1$ now represents the value of the running variable, and $O_2$ represents the outcome variable. See p169 of {cite:t}`reichardt2019quasi`. This will make more sense if you consider the design notation alongside one of the example notebooks.
+
+|     |    |   |    |
+|-----|----|---|----|
+| C: | $O_1$ | X | $O_2$ |
+| C: | $O_1$ |   | $O_2$ |
+
+From an analysis perspective, regression discontinuity designs are very similar to interrupted time series designs. The key difference is that treatment is determined by a cutoff point along a running variable, rather than by time.
+
+## Summary
+This page has offered a brief overview of the tabular notation used to describe quasi-experimental designs. The notation is a useful tool for summarizing the design of a study, and can be used to help identify the strengths and limitations of a study design. But readers are strongly encouraged to consult the original sources when assessing the relative strengths and limitations of making causal claims under different quasi-experimental designs.
+
+## References
+:::{bibliography}
+:filter: docname in docnames
+:::

docs/source/glossary.rst

@@ -18,6 +18,9 @@ Glossary
    Change score analysis
       A statistical procedure where the outcome variable is the difference between the posttest and protest scores.
 
+   Causal impact
+      An umbrella term for the estimated effect of a treatment on an outcome.
+
    Comparative interrupted time-series
    CITS
       An interrupted time series design with added comparison time series observations.
@@ -36,7 +39,6 @@ Glossary
    ITS
       A quasi-experimental design to estimate a treatment effect where a series of observations are collected before and after a treatment. No control group is present.
 
-
    Instrumental Variable regression
    IV
       A quasi-experimental design  to estimate a treatment effect where the is a risk of confounding between the treatment and the outcome due to endogeniety.
@@ -46,7 +48,7 @@ Glossary
 
    Non-equivalent group designs
    NEGD
-      A quasi-experimental design where units are assigned to conditions non-randomly, and not according to a running variable (see Regression discontinuity design).
+      A quasi-experimental design where units are assigned to conditions non-randomly, and not according to a running variable (see Regression discontinuity design). This can be problematic when assigning causal influence of the treatment - differences in outcomes between groups could be due to the treatment or due to differences in the group attributes themselves.
 
    One-group posttest-only design
       A design where a single group is exposed to a treatment and assessed on an outcome measure. There is no pretest measure or comparison group.
@@ -67,11 +69,15 @@ Glossary
       An emprical comparison used to estimate the effects of treatments where units are assigned to treatment conditions randomly.
 
    Regression discontinuity design
+   RDD
       A quasi–experimental comparison to estimate a treatment effect where units are assigned to treatment conditions based on a cut-off score on a quantitative assignment variable (aka running variable).
 
    Regression kink design
       A quasi-experimental research design that estimates treatment effects by analyzing the impact of a treatment or intervention precisely at a defined threshold or "kink" point in a quantitative assignment variable (running variable). Unlike traditional regression discontinuity designs, regression kink design looks for a change in the slope of an outcome variable at the kink, instead of a discontinuity. This is useful when the assignment variable is not discrete, jumping from 0 to 1 at a threshold. Instead, regression kink designs are appropriate when there is a change in the first derivative of the assignment function at the kink point.
 
+   Running variable
+      In regression discontinuity designs, the running variable is the variable that determines the assignment of units to treatment or control conditions. This is typically a continuous variable. Examples could include a test score, age, income, or spatial location. But the running variable would not be time, which is the case in interrupted time series designs.
+
    Sharp regression discontinuity design
       A Regression discontinuity design where allocation to treatment or control is determined by a sharp threshold / step function.
 
@@ -88,7 +94,6 @@ Glossary
    Wilkinson notation
       A notation for describing statistical models :footcite:p:`wilkinson1973symbolic`.
 
-
    Two Stage Least Squares
    2SLS
       An estimation technique for estimating the parameters of an IV regression. It takes its name from the fact that it uses two OLS regressions - a first and second stage.

docs/source/index.rst

@@ -138,9 +138,10 @@ Documentation outline
 =====================
 
 .. toctree::
-   :titlesonly:
+   :caption: Knowledge Base
 
-   glossary
+   design_notation.md
+   glossary.rst
 
 .. toctree::
    :caption: Examples

docs/source/references.bib

@@ -68,3 +68,11 @@ @article{acemoglu2001colonial
   pages={1369--1401},
   year={2001}
 }
+
+@book{shadish_cook_cambell_2002,
+  title={Experimental and quasi-experimental designs for generalized causal inference},
+  author={Cook, Thomas D and Campbell, Donald Thomas and Shadish, William},
+  volume={1195},
+  year={2002},
+  publisher={Houghton Mifflin Boston, MA}
+}