--- title: "Randomization 2: Some common types of designs | *Randomisation 2 : Quelques types courants de conception*" author: "Yannick + Macartan; Alyssa + Vin" date: today bibliography: assets/learningdays-book.bib format: revealjs: embed-resources: true --- ## Some common designs | *Quelques types courants de conception* ::: {.columns} ::: {.column .lang-en width="50%"} 1. Factorial 2. Waitlist (delayed access) 3. Encouragement ::: ::: {.column .lang-fr width="50%"} 1. Factorielle 2. Liste d'attente (accès graduel) 3. Incitations ::: ::: ## 1. Factorial Design | *Conception factorielle* ::: {.columns} ::: {.column .lang-en width="50%"} - In a factorial design, there are two or more **factors** and each factor has two or more conditions. - Each unit is assigned to one of the possible combinations of these conditions. ::: ::: {.column .lang-fr width="50%"} - Dans un plan factoriel, il y a au moins deux **facteurs** et chaque facteur comporte au moins deux conditions. - Chaque unité est assignée à l'une des combinaisons possibles de ces conditions. ::: ::: ## 1. Factorial Design | *Conception factorielle* {.compact-table} | | Transport: No | Transport: Yes | |:--|:--|:--| | **Information: No** | Neither | Transport only | | **Information: Yes** | Information only | Information + Transportation | ## 1. Factorial Design | *Conception factorielle* {.compact-table} | | Transport: No | Transport: Yes | |:--|:--|:--| | **Information: No** | Neither | Transport only | | **Information: Yes** | Information only | Information + Transportation | ::: {.columns} ::: {.column .lang-en width="50%"} There are many possible treatment effects (comparisons) in a factorial design: 1. Conditional Average Treatment Effect (CATE): the ATE of one factor, for a given level of the other factor. ::: ::: {.column .lang-fr width="50%"} Il y a plusieurs effets de traitement (comparaisons) dans une conception factorielle : 1. Effet moyen de traitement conditionnel (CATE) : l'effet moyen d'un des facteurs, en maintenant fixe la valeur de l'autre facteur. ::: ::: ## 1. Factorial Design | *Conception factorielle* {.compact-table} | | Transport: No | Transport: Yes | |:--|:--|:--| | **Information: No** | Neither | Transport only | | **Information: Yes** | Information only | Information + Transportation | ::: {.columns} ::: {.column .lang-en width="50%"} - There are four possible CATEs in this design. - One is the CATE of information conditional on having transport. It compares the cell with information + transport to the cell with transport only. We can ignore the first column. ::: ::: {.column .lang-fr width="50%"} - Il y a 4 CATE possibles dans cette conception. - L'un d'eux est le CATE de l'information sous la condition d'avoir reçu le transport. Cela compare la cellule avec information + transport à la cellule avec transport seulement. Nous ignorons la première colonne. ::: ::: ## 1. Factorial Design | *Conception factorielle* {.compact-table} | | Transport: No | Transport: Yes | |:--|:--|:--| | **Information: No** | Neither | Transport only | | **Information: Yes** | Information only | Information + Transportation | ::: {.columns} ::: {.column .lang-en width="50%"} 2. Interaction effect: the ATE of one factor may differ by levels of the other factor. A treatment effect may be larger or smaller depending on the other treatment. ::: ::: {.column .lang-fr width="50%"} 2. Effet d'interaction : l'effet d'un facteur peut dépendre de la condition d'assignation de l'unité à un autre facteur. Un facteur peut amplifier ou réduire l'effet de l'autre. ::: ::: ## 1. Factorial Design | *Conception factorielle* {.compact-table} | | Transport: No | Transport: Yes | |:--|:--|:--| | **Information: No** | Neither | Transport only | | **Information: Yes** | Information only | Information + Transportation | ::: {.columns} ::: {.column .lang-en width="50%"} - Does having transport change the effect of information? We compare the CATE of information with transport (from before) to the CATE of information without transport. - If the 2 CATEs are different, we say there is an interaction effect. ::: ::: {.column .lang-fr width="50%"} - Est-ce que le transport change l'effet de l'information ? Nous comparons le CATE de l'information avec transport (comme avant) au CATE de l'information sans transport. - Si les 2 CATE sont différents, on dira qu'il y a un effet d'interaction. ::: ::: ## 1. Factorial Design | *Conception factorielle* {.compact-table} | | Transport: No | Transport: Yes | |:--|:--|:--| | **Information: No** | Neither | Transport only | | **Information: Yes** | Information only | Information + Transportation | ::: {.columns} ::: {.column .lang-en width="50%"} 3. Average effect of one treatment (given the distribution of the other treatment). ::: ::: {.column .lang-fr width="50%"} 3. Effet moyen d'un traitement (étant donné la distribution de l'autre traitement). ::: ::: ## 1. Factorial Design | *Conception factorielle* {.smaller} ### Design-based analysis ::: {.columns} ::: {.column .lang-en width="50%"} Main estimands ::: ::: {.column .lang-fr width="50%"} Paramètres ::: ::: ```{r factorial-estimands-table, echo = FALSE, results = "asis"} library(kableExtra) grey <- "#E8E8E8" blue <- "#DCEEFB" green <- "#D4EDDA" factorial_estimands_tab <- data.frame( v1 = c("$X_1$", "", "", ""), v2 = c("0", "1", "All", "Diff"), v3 = c( "$\\overline{y}_{00}$", "$\\overline{y}_{10}$", "$\\overline{y}_{.0}$", "$\\tau_1 \\mid X_2{=}0$" ), v4 = c( "$\\overline{y}_{01}$", "$\\overline{y}_{11}$", "$\\overline{y}_{.1}$", "$\\tau_1 \\mid X_2{=}1$" ), v5 = c("$\\overline{y}_{0.}$", "$\\overline{y}_{1.}$", "$\\overline{y}$", "$\\tau_1$"), v6 = c( "$\\tau_2 \\mid X_1{=}0$", "$\\tau_2 \\mid X_1{=}1$", "$\\tau_2$", "$\\tau_{12}$" ), check.names = FALSE ) cat( factorial_estimands_tab |> kable( col.names = c("", "", "0", "1", "All", "Diff"), escape = FALSE, align = "c" ) |> add_header_above(c(" " = 2, "$X_2$" = 3, " " = 1)) |> column_spec(3, background = c(grey, grey, blue, green)) |> column_spec(4, background = c(grey, grey, blue, green)) |> column_spec(5, background = c(blue, blue, blue, green)) |> column_spec(6, background = rep(green, 4)) |> kable_styling(full_width = FALSE, font_size = 18, position = "center") ) ``` ## 1. Factorial Design | *Conception factorielle* ```{r, echo = TRUE} library(randomizr) set.seed(12345) information <- complete_ra(N = 24, m = 12) transport <- block_ra(blocks = information) table(information, transport) ``` ## 2. Waitlist design (delayed access) | *Liste d'attente (accès graduel)* ::: {.columns} ::: {.column .lang-en width="50%"} - Situation: Only a certain number of units can be treated at a time. Once treated, a unit stays in treatment. - When an intervention can be or must be rolled out in stages, you can randomize the order (*timing*) in which units are treated. ::: ::: {.column .lang-fr width="50%"} - Situation : seul un certain nombre d'unités peuvent être traitées en même temps. Une fois traitée, une unité reste en traitement. - Lorsqu'une intervention peut ou doit être déployée par étapes, vous pouvez randomiser l'ordre (*timing*) de traitement des unités. ::: ::: ## 2. Waitlist design (delayed access) | *Liste d'attente (accès graduel)* ![](images/delayed_access.png){width="300px" fig-align="center"} ::: {.columns} ::: {.column .lang-en width="50%"} - Your control group are the as-yet untreated units. ::: ::: {.column .lang-fr width="50%"} - Votre groupe de contrôle est constitué des unités pas encore traitées. ::: ::: ## 2. Waitlist design (delayed access) | *Liste d'attente (accès graduel)* ::: {.columns} ::: {.column .lang-en width="50%"} - We need to assume **no anticipation**. - This means that the potential outcome is not affected by future treatment status. ::: ::: {.column .lang-fr width="50%"} - Nous devons faire l'hypothèse d'**aucune anticipation**. - Cela signifie que le résultat potentiel n'est pas affecté par l'état futur du traitement. ::: ::: ## 2. Waitlist design (delayed access) | *Liste d'attente (accès graduel)* ::: {.columns} ::: {.column .lang-en width="50%"} - We analyze the data from all time periods together. - Be careful: the probability of assignment to treatment will vary over time because units assigned to treatment in earlier stages are not eligible for treatment in later stages. ::: ::: {.column .lang-fr width="50%"} - Nous analysons les données de toutes les périodes ensemble. - Attention : la probabilité d'assignation au traitement variera dans le temps car les unités assignées au traitement à des stades antérieurs ne sont plus éligibles aux stades ultérieurs. ::: ::: # 3. Encouragement design | *Conception incitative* ## 3. Encouragement design | *Conception incitative* ```{r, include = FALSE} library(CausalQueries) ``` ::: {.columns} ::: {.column .lang-en width="50%"} ```{r, echo = FALSE} make_model("Z -> D -> Y <- D") |> plot_model(labels = c("Z (encouragement)", "D (treatment)", "Y (outcome)"), nodecol = "white", textcol = "black", textsize = 8) ``` ::: ::: {.column .lang-fr width="50%"} ```{r, echo = FALSE} make_model("Z -> D -> Y <- D") |> plot_model(labels = c("Z (incitation)", "D (traitement)", "Y (résultat)"), nodecol = "white", textcol = "black", textsize = 8) ``` ::: ::: ## 3. Encouragement design | *Conception incitative* ::: {.columns} ::: {.column .lang-en width="50%"} - Situation: You can't force people to take (receive) your treatment. Treatment assigned is not the same as treatment received. - We can randomize **encouragement** $Z$ to take the treatment, such as a request to drink coffee or offering a subsidy to participate in a program. - We measure the encouragement $Z$, taking the treatment $D$, and the outcome $Y$. ::: ::: {.column .lang-fr width="50%"} - Situation : vous ne pouvez pas forcer les gens à prendre (recevoir) le traitement. Le traitement assigné n'est pas le même que le traitement reçu. - Nous pouvons randomiser l'**incitation** $Z$ à suivre le traitement, par exemple en demandant de boire un café ou en offrant une subvention. - On mesure l'incitation $Z$, le traitement reçu $D$, et le résultat $Y$. ::: ::: ## 3. Encouragement design | *Conception incitative* ::: {.columns} ::: {.column .lang-en width="50%"} - $D$ is not randomized! - So it is difficult to learn the ATE of $D$ on $Y$. - But we may be able to target other estimands. ::: ::: {.column .lang-fr width="50%"} - $D$ n'est pas randomisé ! - Il est alors difficile d'estimer l'ATE de $D$ sur $Y$. - Mais nous pouvons peut-être cibler d'autres paramètres. ::: ::: ## 3. Encouragement design | *Conception incitative* ::: {.columns} ::: {.column .lang-en width="50%"} **Estimand 1:** - We can learn the average effect of the *encouragement* $Z$ to take the treatment on the outcome $Y$. - This is the ATE of $Z$, also known as **ITT**, the intent-to-treat effect. ::: ::: {.column .lang-fr width="50%"} **Paramètre 1 :** - Nous pouvons estimer l'effet moyen de *l'encouragement* $Z$ à suivre le traitement sur le résultat $Y$. - C'est l'ATE de $Z$, aussi appelé **ITT**, l'effet de l'intention de traiter. ::: ::: ## 3. Encouragement design | *Conception incitative* ::: {.columns} ::: {.column .lang-en width="50%"} **Estimand 2:** - With some additional assumptions, we can also learn the average effect of *taking the treatment* $D$ for Compliers. - This is known as the Complier Average Causal Effect (CACE) or Local Average Treatment Effect (LATE). ::: ::: {.column .lang-fr width="50%"} **Paramètre 2 :** - Avec quelques hypothèses supplémentaires, nous pouvons aussi estimer l'effet moyen de *l'acceptation du traitement* $D$ pour les conformistes. - C'est l'effet causal moyen du conformiste (CACE) ou l'effet moyen local du traitement (LATE). ::: ::: ## 3. Encouragement design | *Conception incitative* ::: {.columns} ::: {.column .lang-en width="50%"} - Who are Compliers? They are units that would take the treatment if encouraged ($D(1)=1$) and not if not ($D(0)=0$). ::: ::: {.column .lang-fr width="50%"} - Qui sont les conformistes ? Des unités qui prendraient le traitement si encouragées ($D(1)=1$) et pas sinon ($D(0)=0$). ::: ::: | Type | $Z=1$ | $Z=0$ | |:--|:--:|:--:| | Always Taker | $D(1)=1$ | $D(0)=1$ | | **Complier** | $D(1)=1$ | $D(0)=0$ | | Never Taker | $D(1)=0$ | $D(0)=0$ | | Defier | $D(1)=0$ | $D(0)=1$ | ## 3. Encouragement design | *Conception incitative* ::: {.columns} ::: {.column .lang-en width="50%"} - Additional assumption 1: For CACE, we need **excludability** (exclusion restriction): the encouragement $Z$ only affects the outcome $Y$ through taking the treatment $D$. ::: ::: {.column .lang-fr width="50%"} - Hypothèse supplémentaire 1 : pour le CACE, nous avons besoin de **l'excluabilité** (restriction d'exclusion) : l'encouragement $Z$ n'affecte $Y$ qu'à travers le traitement $D$. ::: ::: ## 3. Encouragement design | *Conception incitative* ::: {.columns} ::: {.column .lang-en width="50%"} - Additional assumption 2: For CACE using this design, we also need the assumption of **monotonicity**. This means no Defiers. ::: ::: {.column .lang-fr width="50%"} - Hypothèse supplémentaire 2 : pour le CACE, nous avons aussi besoin de **monotonicité**. Cela signifie qu'il n'y a pas de non-conformistes. ::: ::: | Type | $Z=1$ | $Z=0$ | |:--|:--:|:--:| | Always Taker | $D(1)=1$ | $D(0)=1$ | | **Complier** | $D(1)=1$ | $D(0)=0$ | | Never Taker | $D(1)=0$ | $D(0)=0$ | ## 3. Encouragement design | *Conception incitative* ::: {.columns} ::: {.column .lang-en width="50%"} - Careful (1)! Do not compare those who take treatment ($D=1$) to those who do not ($D=0$). Taking treatment is not randomly assigned and the two groups are not comparable. ::: ::: {.column .lang-fr width="50%"} - Attention (1) ! Ne comparez pas les sujets qui prennent le traitement ($D=1$) à ceux qui ne le prennent pas ($D=0$). La prise du traitement n'est pas aléatoire et les deux groupes ne sont pas comparables. ::: ::: ## 3. Encouragement design | *Conception incitative* ```{r, include = FALSE} library(ggplot2) encouragement_type_dag <- function(labels) { labs <- data.frame( x = c(1.1, 4.5, 4.5, 4.5), y = c(3.0, 5.0, 3.0, 1.0), label = labels ) arrows <- data.frame( x = c(4.5, 2.0, 1.6, 4.5), y = c(4.55, 3.0, 2.65, 2.45), xend = c(4.5, 3.55, 3.85, 4.5), yend = c(3.45, 3.0, 1.35, 1.55) ) ggplot() + geom_segment( data = arrows, aes(x = x, y = y, xend = xend, yend = yend), arrow = arrow(length = unit(0.15, "cm"), type = "closed"), linewidth = 0.45, inherit.aes = FALSE ) + geom_text( data = labs, aes(x = x, y = y, label = label), size = 4.2, inherit.aes = FALSE ) + coord_fixed(ratio = 0.9, xlim = c(0, 6), ylim = c(0, 6), expand = FALSE) + theme_void() } ``` ::: {.columns} ::: {.column .lang-en width="50%"} ```{r, echo = FALSE, fig.width = 5, fig.height = 4} encouragement_type_dag(c("Type", "Z (encouragement)", "D (treatment)", "Y (outcome)")) ``` ::: ::: {.column .lang-fr width="50%"} ```{r, echo = FALSE, fig.width = 5, fig.height = 4} encouragement_type_dag(c("Type", "Z (incitation)", "D (traitement)", "Y (résultat)")) ``` ::: ::: ## 3. Encouragement design | *Conception incitative* ::: {.columns} ::: {.column .lang-en width="50%"} - Careful (2)! For the CACE, we have to assess whether excludability and monotonicity hold in our study. - If not, we can still target the ITT with the other standard key assumptions. ::: ::: {.column .lang-fr width="50%"} - Attention (2) ! Il faut évaluer si la restriction d'exclusion et la monotonicité sont raisonnables dans notre étude. - Sinon, nous pouvons encore cibler l'ITT avec les autres hypothèses clés standardes. ::: ::: ## 3. Encouragement design | *Conception incitative* ::: {.columns} ::: {.column .lang-en width="50%"} - When would you target ITT? When would you target CACE? - ITT: Policy can directly change $Z$, but not $D$. - CACE: We may want to know the effect of $D$ even if it's for just some of the units. ::: ::: {.column .lang-fr width="50%"} - Quand cibler l'ITT ? Quand cibleriez-vous le CACE ? - ITT : la politique peut modifier directement $Z$, mais pas $D$. - CACE : nous pourrions vouloir connaître l'effet de $D$ même si ce n'est que pour certaines unités. ::: ::: ## Summary | *Résumé* ::: {.columns} ::: {.column .lang-en width="50%"} - Factorial: 2 or more treatments with possible interaction - Waitlist: when we have a constraint on how many treatments can be delivered at once - Encouragement: when we can't force units to take the treatment ::: ::: {.column .lang-fr width="50%"} - Factorielle : 2 traitements ou plus avec interaction possible - Liste d'attente : contrainte sur le nombre de traitements délivrés à la fois - Incitation : quand on ne peut pas obliger les unités à recevoir le traitement ::: ::: ## Resources | *Ressources* ::: {.columns} ::: {.column .lang-en width="50%"} EGAP Methods Guide on Randomization: ::: ::: {.column .lang-fr width="50%"} Guide des méthodes EGAP sur la randomisation : ::: :::