class: center, middle, inverse, title-slide # <b>Analysing Factorial Designs </b> ## Data Analysis for Psychology in R 2<br><br> ### Tom Booth and Alex Doumas ### Department of Psychology<br>The University of Edinburgh ### AY 2020-2021 --- # Weeks Learning Objectives 1. Interpret the output from a model using dummy coding and sum-to-zero coding. 2. Create specific contrast matrices to test specific effects. 3. Recognise other forms of contrasts. 4. Construct models to test factorial designs. --- # Topics for today + Tabulating data from factorial design. + Recap factorial designs effects of interest. + Main effects + Simple effects/contrasts + Interactions + Show the tests of main effects via model comparison using `\(F\)`-tests. --- # Example + The data comes from a study into patient care in a paediatric wards. + A researcher was interested in whether the subjective well-being of patients differed dependent on the post-operation treatment schedule they were given, and the hospital in which they were staying. + **Condition 1**: `Treatment` (Levels: TreatA, TreatB, TreatC. + **Condition 2**: `Hosp` (Levels: Hosp1, Hosp2). + Total sample n = 180 (30 patients in each of 6 groups). + Between person design. + **Outcome**: Subjective well-being (SWB) + An average of multiple raters (the patient, a member of their family, and a friend). + SWB score ranged from 0 to 20. --- # The data ```r hosp_tbl <- read_csv("hospital.csv", col_types = "dff") hosp_tbl %>% slice(1:10) ``` ``` ## # A tibble: 10 x 3 ## SWB Treatment Hospital ## <dbl> <fct> <fct> ## 1 6.2 TreatA Hosp1 ## 2 15.9 TreatA Hosp1 ## 3 7.2 TreatA Hosp1 ## 4 11.3 TreatA Hosp1 ## 5 11.2 TreatA Hosp1 ## 6 9 TreatA Hosp1 ## 7 14.5 TreatA Hosp1 ## 8 7.3 TreatA Hosp1 ## 9 13.7 TreatA Hosp1 ## 10 12.6 TreatA Hosp1 ``` --- # Table of means .pull-left[ ```r mean(hosp_tbl$SWB) ``` ``` ## [1] 9.880556 ``` ```r aggregate(SWB ~ Treatment + Hospital, hosp_tbl, mean) ``` ``` ## Treatment Hospital SWB ## 1 TreatA Hosp1 10.800000 ## 2 TreatB Hosp1 9.430000 ## 3 TreatC Hosp1 10.103333 ## 4 TreatA Hosp2 7.853333 ## 5 TreatB Hosp2 13.116667 ## 6 TreatC Hosp2 7.980000 ``` ] .pull-right[ ```r aggregate(SWB ~ Hospital, hosp_tbl, mean) ``` ``` ## Hospital SWB ## 1 Hosp1 10.11111 ## 2 Hosp2 9.65000 ``` ```r aggregate(SWB ~ Treatment, hosp_tbl, mean) ``` ``` ## Treatment SWB ## 1 TreatA 9.326667 ## 2 TreatB 11.273333 ## 3 TreatC 9.041667 ``` ] --- # Table of means + All of the above gives us a full table of means <table> <thead> <tr> <th style="text-align:left;"> </th> <th style="text-align:left;"> Hosp1 </th> <th style="text-align:left;"> Hosp2 </th> <th style="text-align:left;"> Marginal </th> </tr> </thead> <tbody> <tr> <td style="text-align:left;"> TreatA </td> <td style="text-align:left;"> 10.80 </td> <td style="text-align:left;"> 7.85 </td> <td style="text-align:left;"> 9.33 </td> </tr> <tr> <td style="text-align:left;"> TreatB </td> <td style="text-align:left;"> 9.43 </td> <td style="text-align:left;"> 13.11 </td> <td style="text-align:left;"> 11.27 </td> </tr> <tr> <td style="text-align:left;"> TreatC </td> <td style="text-align:left;"> 10.10 </td> <td style="text-align:left;"> 7.98 </td> <td style="text-align:left;"> 9.04 </td> </tr> <tr> <td style="text-align:left;"> Marginal </td> <td style="text-align:left;"> 10.11 </td> <td style="text-align:left;"> 9.65 </td> <td style="text-align:left;"> 9.88 </td> </tr> </tbody> </table> --- # Hypotheses we test in Factorial Designs + Main effects + An overall, or average, effect of a condition. + In our example, is there an effect of `Treatment` ignoring `Hospital` (and vice versa)? + Simple contrasts/effects + An effect of one condition at a specific level of another. + Is there an effect of `Hospital` for those receiving `Treatment A`? (...and so on for all combinations.) + Interactions (categorical*categorical) + A change in the effect of some condition as a function of another. + Does the effect of `Treatment` differ by `Hospital`? --- # Our model and coefficients + The linear model with two categorical variables: `$$y_{ijk} = b_0 + \alpha_i + \tau_j + \epsilon_{ijk}$$` + where; + i = 1 .... g_A, j = 1 ... g_B, k = 1... n + `\(y_{ijk}\)` is the kth observation of level i of the first factor and level j of the second factor, + `\(\alpha_i\)` is the effect of the level i of the first factor, + `\(\tau_j\)` is the effect of level j of the second factor. + But remember whichever coding scheme we use, we have `\(g\)`-1 variables representing the condition. + So for `Treatment` we have 2 predictors (D1 & D2) + And for `Hospital` we have 1 predictor (D3) + We can write the linear model more explicitly as: `$$y_{ijk} = b_0 + \underbrace{(b_1D_1 + b_2D_2)}_{\text{Treatment}} + \underbrace{b_3D_3}_{\text{Hospital}} + \epsilon_{i}$$` --- # Number of interaction terms + To include terms for the interaction, we need to cross each level of one condition with the levels of the other. + In general this requirement will mean we need ( `\(r\)`-1)( `\(c\)`-1) interaction terms + where `\(c\)` and `\(r\)` represent the number of levels of each condition. + In our case this is (3-1)(2-1) = 2 `$$y_{ijk} = b_0 + \underbrace{(b_1D_1 + b_2D_2)}_{\text{Treatment}} + \underbrace{b_3D_3}_{\text{Hospital}} + \underbrace{b_4D_{13} + b_5D_{23}}_{\text{Interactions}} + \epsilon_{i}$$` + We will talk get into more detail about this practice soon. --- # Testing the overall effects + The goal of our `\(F\)`-tests for the overall effect of a condition or interaction, is to assess whether models which include all coefficients that code the condition improve the model. + Hopefully, this practice sounds familiar to you. + It's just using incremental `\(F\)` tests. + To do incremental `\(F\)` tests, we need to define a set of models: ```r m1 <- lm(SWB ~ Treatment, data = hosp_tbl) m2 <- lm(SWB ~ Hospital, data = hosp_tbl) m3 <- lm(SWB ~ Treatment + Hospital, data = hosp_tbl) m4 <- lm(SWB ~ Treatment + Hospital + Treatment*Hospital, data = hosp_tbl) ``` --- # Testing the overall effects + For the effect of `Treatment`: ```r m2 <- lm(SWB ~ Hospital, data = hosp_tbl) m3 <- lm(SWB ~ Treatment + Hospital, data = hosp_tbl) anova(m2,m3) ``` ``` ## Analysis of Variance Table ## ## Model 1: SWB ~ Hospital ## Model 2: SWB ~ Treatment + Hospital ## Res.Df RSS Df Sum of Sq F Pr(>F) ## 1 178 1283.5 ## 2 176 1106.5 2 177.02 14.078 2.13e-06 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` + An effect of Treatment --- # Testing the overall effects + For the effect of `Hospital`: ```r m1 <- lm(SWB ~ Treatment, data = hosp_tbl) m3 <- lm(SWB ~ Treatment + Hospital, data = hosp_tbl) anova(m1,m3) ``` ``` ## Analysis of Variance Table ## ## Model 1: SWB ~ Treatment ## Model 2: SWB ~ Treatment + Hospital ## Res.Df RSS Df Sum of Sq F Pr(>F) ## 1 177 1116.1 ## 2 176 1106.5 1 9.5681 1.5219 0.219 ``` + No effect of hospital --- # Testing the overall effects + For the effect of interaction: ```r m3 <- lm(SWB ~ Treatment + Hospital, data = hosp_tbl) m4 <- lm(SWB ~ Treatment + Hospital + Treatment*Hospital, data = hosp_tbl) anova(m3,m4) ``` ``` ## Analysis of Variance Table ## ## Model 1: SWB ~ Treatment + Hospital ## Model 2: SWB ~ Treatment + Hospital + Treatment * Hospital ## Res.Df RSS Df Sum of Sq F Pr(>F) ## 1 176 1106.51 ## 2 174 714.34 2 392.18 47.764 < 2.2e-16 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` + An interaction --- # Testing the overall effects + Using `anova()`: ```r m4 <- lm(SWB ~ Treatment + Hospital + Treatment*Hospital, data = hosp_tbl) anova(m4) ``` ``` ## Analysis of Variance Table ## ## Response: SWB ## Df Sum Sq Mean Sq F value Pr(>F) ## Treatment 2 177.02 88.511 21.5597 4.315e-09 *** ## Hospital 1 9.57 9.568 2.3306 0.1287 ## Treatment:Hospital 2 392.18 196.088 47.7635 < 2.2e-16 *** ## Residuals 174 714.34 4.105 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ``` + The values are not identical (there are some devils in detail), but you can see the pattern of results is the same in both approaches. --- # Summary of today + Look at constructing `\(F\)`-tests for the overall effect of conditions (categorical variables) from a factorial design. + Now we can move on to consider the interaction term in more detail.