class: center, middle, inverse, title-slide # IDS 702: Module 4.5 ## Multilevel/hierarchical logistic models ### Dr. Olanrewaju Michael Akande --- ## Multilevel linear models - The same idea and approach used to build multilevel models for normal data can be used to build multilevel logistic (and probit) models for binary outcomes. -- - Recall that for a .hlight[varying-intercepts linear model] with one individual-level predictor, we have .block[ .small[ $$ `\begin{split} y_{ij} & = \beta_{0} + \gamma_{0j} + \beta_1 x_{1ij} + \epsilon_{ij}; \ \ \ i = 1, \ldots, n_j; \ \ \ j = 1, \ldots, J; \\ \epsilon_{ij} & \sim N(0, \sigma^2); \\ \gamma_{0j} & \sim N(0, \tau_0^2) \end{split}` $$ ] ] where `\(x_{1ij}\)` can be replaced with `\(x_{1j}\)` for a group-level predictor. -- - This model is a compromise between complete pooling across groups of a grouping variable, such as counties in the radon example for last class (that is, same intercept for each county), and no pooling (estimating a separate intercept for each county without borrowing information). -- - The degree of pooling is determined by the amount of information within and between groups. --- ## Multilevel logistic models - We can use the same idea to build a .hlight[varying-intercepts logistic model]. - That is, .block[ .small[ $$ `\begin{split} y_{ij} | x_{ij} & \sim \textrm{Bernoulli}(\pi_{ij}); \ \ \ i = 1, \ldots, n_j; \ \ \ j = 1, \ldots, J; \\ \textrm{log}\left(\dfrac{\pi_{ij}}{1-\pi_{ij}}\right) & = \beta_{0} + \gamma_{0j} + \beta_1 x_{1ij}; \\ \gamma_{0j} & \sim N(0, \sigma_0^2) \end{split}` $$ ] ] where once again, `\(x_{1ij}\)` is an .hlight[individual-level predictor] which can be replaced with `\(x_{1j}\)` for a .hlight[group-level predictor]. -- - The Gelman and Hill book uses the following notation instead .block[ .small[ $$ `\begin{split} y_i | x_i & \sim \textrm{Bernoulli}(\pi_i); \ \ \ i = 1, \ldots, n; \ \ \ j = 1, \ldots, J; \\ \textrm{log}\left(\dfrac{\pi_i}{1-\pi_i}\right) & = \beta_{0} + \gamma_{0j[i]} + \beta_1 x_{i1}; \\ \gamma_{0j} & \sim N(0, \sigma_0^2). \end{split}` $$ ] ] -- - .hlight[I will use this notation in this module and the next]. --- ## Varying-intercepts logistic model Inverse logit functions for varying-intercepts logistic models. <img src="4-5-hierarchical-logistic-models_files/figure-html/unnamed-chunk-1-1.png" style="display: block; margin: auto;" /> --- ## Multilevel logistic models - It is easy to extend this model to allow for .hlight[varying-slopes] or both .hlight[varying-intercepts and varying-slopes] just like we had for multilevel linear models. -- - The interpretations of the fixed effect(s) in multilevel logistic models follow directly from what we had for the standard logistic models, that is, log-odds, odds and odds ratios. -- - The only difference now is the hierarchy in our data which allows us to borrow information across groups. -- - One way to think about this is that we expect odds and odd-ratios to be more similar for observations within the same group, but we allow for some similarity across groups via partial pooling. --- ## Varying-slopes logistic model Inverse logit functions for varying-slopes logistic models. <img src="4-5-hierarchical-logistic-models_files/figure-html/unnamed-chunk-2-1.png" style="display: block; margin: auto;" /> --- ## Varying-intercepts, varying-slopes logistic model Inverse logit functions for varying-intercepts, varying-slopes logistic models. <img src="4-5-hierarchical-logistic-models_files/figure-html/unnamed-chunk-3-1.png" style="display: block; margin: auto;" /> --- ## 1988 elections analysis - To illustrate how to fit and interpret the results of multilevel logistic models, we will use a sample data on election polls. -- - National opinion polls are conducted by a variety of organizations (e.g., media, polling organizations, campaigns) leading up to elections. -- - While many of the best opinion polls are conducted at a national level, it can also be often interesting to estimate voting opinions and preferences at the state or even local level. -- - Well-designed polls are generally based on national random samples with corrections for nonresponse based on a variety of demographic factors (e.g., sex, ethnicity, race, age, education). -- - The data is from CBS News surveys conducted during the week before the 1988 election. -- - Respondents were asked about their preferences for either the Republican candidate (Bush Sr.) or the Democratic candidate (Dukakis). --- ## 1988 elections analysis The dataset includes 2193 observations from one of eight surveys (the most recent CBS News survey right before the election) in the original full data. .small[ Variable | Description :------------- | :------------ org | cbsnyt = CBS/NYT .hlight[bush] | .hlight[1 = preference for Bush Sr., 0 = otherwise] state | 1-51: 50 states including DC (number 9) edu | education: 1=No HS, 2=HS, 3=Some College, 4=College Grad age | 1=18-29, 2=30-44, 3=45-64, 4=65+ female | 1=female, 0=male black | 1=black, 0=otherwise region | 1=NE, 2=S, 3=N, 4=W, 5=DC v_prev | average Republican vote share in the three previous elections (adjusted for home-state and home-region effects in the previous elections) ] -- Given that the data has a natural multilevel structure (through `state` and `region`), it makes sense to explore multilevel models for this data. -- We will do just that in the next module. --- ## 1988 elections analysis - Both voting turnout and preferences often depend on a complex combination of demographic factors. -- - In our example dataset, we have demographic factors such as biological sex, race, age, education, which we may all want to look at by state, resulting in `\(2 \times 2 \times 4 \times 4 \times 51 = 3264\)` potential categories of respondents. -- - We may even want to control for `region`, adding to the number of categories. -- - Clearly, without a very large survey (most political survey poll around 1000 people), we will need to make assumptions in order to even obtain estimates in each category. -- - We usually cannot include all interactions; we should therefore select those to explore (through EDA and background knowledge). -- - The data is in the file `polls_subset.txt` on Sakai. --- ## 1988 elections analysis ```r ###### Load the data polls_subset <- read.table("data/polls_subset.txt",header=TRUE) str(polls_subset) ``` ``` ## 'data.frame': 2193 obs. of 10 variables: ## $ org : chr "cbsnyt" "cbsnyt" "cbsnyt" "cbsnyt" ... ## $ survey: int 9158 9158 9158 9158 9158 9158 9158 9158 9158 9158 ... ## $ bush : int NA 1 0 0 1 1 1 1 0 0 ... ## $ state : int 7 39 31 7 33 33 39 20 33 40 ... ## $ edu : int 3 4 2 3 2 4 2 2 4 1 ... ## $ age : int 1 2 4 1 2 4 2 4 3 3 ... ## $ female: int 1 1 1 1 1 1 0 1 0 0 ... ## $ black : int 0 0 0 0 0 0 0 0 0 0 ... ## $ region: int 1 1 1 1 1 1 1 1 1 1 ... ## $ v_prev: num 0.567 0.527 0.564 0.567 0.524 ... ``` ```r head(polls_subset) ``` ``` ## org survey bush state edu age female black region v_prev ## 1 cbsnyt 9158 NA 7 3 1 1 0 1 0.5666333 ## 2 cbsnyt 9158 1 39 4 2 1 0 1 0.5265667 ## 3 cbsnyt 9158 0 31 2 4 1 0 1 0.5641667 ## 4 cbsnyt 9158 0 7 3 1 1 0 1 0.5666333 ## 5 cbsnyt 9158 1 33 2 2 1 0 1 0.5243666 ## 6 cbsnyt 9158 1 33 4 4 1 0 1 0.5243666 ``` --- ## 1988 elections analysis ```r summary(polls_subset) ``` ``` ## org survey bush state ## Length:2193 Min. :9158 Min. :0.0000 Min. : 1.00 ## Class :character 1st Qu.:9158 1st Qu.:0.0000 1st Qu.:14.00 ## Mode :character Median :9158 Median :1.0000 Median :26.00 ## Mean :9158 Mean :0.5578 Mean :26.11 ## 3rd Qu.:9158 3rd Qu.:1.0000 3rd Qu.:39.00 ## Max. :9158 Max. :1.0000 Max. :51.00 ## NA's :178 ## edu age female black ## Min. :1.000 Min. :1.000 Min. :0.0000 Min. :0.00000 ## 1st Qu.:2.000 1st Qu.:2.000 1st Qu.:0.0000 1st Qu.:0.00000 ## Median :2.000 Median :2.000 Median :1.0000 Median :0.00000 ## Mean :2.653 Mean :2.289 Mean :0.5887 Mean :0.07615 ## 3rd Qu.:4.000 3rd Qu.:3.000 3rd Qu.:1.0000 3rd Qu.:0.00000 ## Max. :4.000 Max. :4.000 Max. :1.0000 Max. :1.00000 ## ## region v_prev ## Min. :1.000 Min. :0.1530 ## 1st Qu.:2.000 1st Qu.:0.5278 ## Median :2.000 Median :0.5481 ## Mean :2.431 Mean :0.5550 ## 3rd Qu.:3.000 3rd Qu.:0.5830 ## Max. :5.000 Max. :0.6927 ## ``` --- ## 1988 elections analysis ```r polls_subset$v_prev <- polls_subset$v_prev*100 #rescale polls_subset$region_label <- factor(polls_subset$region,levels=1:5, labels=c("NE","S","N","W","DC")) #we consider DC as a separate region due to its distinctive voting patterns polls_subset$edu_label <- factor(polls_subset$edu,levels=1:4, labels=c("No HS","HS","Some College","College Grad")) polls_subset$age_label <- factor(polls_subset$age,levels=1:4, labels=c("18-29","30-44","45-64","65+")) #the data includes states but without the names, which we will need, #so let's grab that from R datasets data(state) #"state" is an R data file (type ?state from the R command window for info) state.abb #does not include DC, so we will create ours ``` ``` ## [1] "AL" "AK" "AZ" "AR" "CA" "CO" "CT" "DE" "FL" "GA" "HI" "ID" "IL" "IN" "IA" ## [16] "KS" "KY" "LA" "ME" "MD" "MA" "MI" "MN" "MS" "MO" "MT" "NE" "NV" "NH" "NJ" ## [31] "NM" "NY" "NC" "ND" "OH" "OK" "OR" "PA" "RI" "SC" "SD" "TN" "TX" "UT" "VT" ## [46] "VA" "WA" "WV" "WI" "WY" ``` ```r #In the polls data, DC is the 9th "state" in alphabetical order state_abbr <- c (state.abb[1:8], "DC", state.abb[9:50]) polls_subset$state_label <- factor(polls_subset$state,levels=1:51,labels=state_abbr) rm(list = ls(pattern = "state")) #remove unnecessary values in the environment ``` --- ## 1988 elections analysis ```r ###### View properties of the data head(polls_subset) ``` ``` ## org survey bush state edu age female black region v_prev region_label ## 1 cbsnyt 9158 NA 7 3 1 1 0 1 56.66333 NE ## 2 cbsnyt 9158 1 39 4 2 1 0 1 52.65667 NE ## 3 cbsnyt 9158 0 31 2 4 1 0 1 56.41667 NE ## 4 cbsnyt 9158 0 7 3 1 1 0 1 56.66333 NE ## 5 cbsnyt 9158 1 33 2 2 1 0 1 52.43666 NE ## 6 cbsnyt 9158 1 33 4 4 1 0 1 52.43666 NE ## edu_label age_label state_label ## 1 Some College 18-29 CT ## 2 College Grad 30-44 PA ## 3 HS 65+ NJ ## 4 Some College 18-29 CT ## 5 HS 30-44 NY ## 6 College Grad 65+ NY ``` ```r dim(polls_subset) ``` ``` ## [1] 2193 14 ``` --- ## 1988 elections analysis ```r ###### View properties of the data str(polls_subset) ``` ``` ## 'data.frame': 2193 obs. of 14 variables: ## $ org : chr "cbsnyt" "cbsnyt" "cbsnyt" "cbsnyt" ... ## $ survey : int 9158 9158 9158 9158 9158 9158 9158 9158 9158 9158 ... ## $ bush : int NA 1 0 0 1 1 1 1 0 0 ... ## $ state : int 7 39 31 7 33 33 39 20 33 40 ... ## $ edu : int 3 4 2 3 2 4 2 2 4 1 ... ## $ age : int 1 2 4 1 2 4 2 4 3 3 ... ## $ female : int 1 1 1 1 1 1 0 1 0 0 ... ## $ black : int 0 0 0 0 0 0 0 0 0 0 ... ## $ region : int 1 1 1 1 1 1 1 1 1 1 ... ## $ v_prev : num 56.7 52.7 56.4 56.7 52.4 ... ## $ region_label: Factor w/ 5 levels "NE","S","N","W",..: 1 1 1 1 1 1 1 1 1 1 ... ## $ edu_label : Factor w/ 4 levels "No HS","HS","Some College",..: 3 4 2 3 2 4 2 2 4 1 ... ## $ age_label : Factor w/ 4 levels "18-29","30-44",..: 1 2 4 1 2 4 2 4 3 3 ... ## $ state_label : Factor w/ 51 levels "AL","AK","AZ",..: 7 39 31 7 33 33 39 20 33 40 ... ``` --- class: center, middle # What's next? ### Move on to the readings for the next module!