Investigating Models with Two or Three Categories

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1 Ronald H. Heck and Lynn N. Tabata 1 Investigating Models with Two or Three Categories For the past few weeks we have been working with discriminant analysis. Let s now see what the same sort of model might look like if we used logistic regression with a dichotomous outcome or multinomial outcome. We could also extend this to the multinomial case (e.g., three job categories), which is similar to the three-group discriminant analyses with which we have been working. We will save that second type of categorical model one for another day, however. The first type of model considers examining a model where the outcome is dichotomous (coded 0 and 1) outcome. When there are only a two categories in the scale of measurement (e.g., such as no or yes, or three categories (such as clerk, custodian, and manager) another approach is to incorporate a necessary transformation and choice of error distribution directly into the statistical modeling approach (Hox, 2010). These types of model are often referred to as generalized linear models (McCullagh & Nelder, 1989). As Hox notes, generalized linear models make it possible to extend standard regression models in several ways including the inclusion of non-normal error distribution and by using nonlinear link functions; that is, a means of linking expected values of the outcome variable (e.g., a binary, multinomial) to an underlying (latent) variable that represents predicted values of the outcome. This is accomplished through what is referred to as a link function. The probability that y takes on a particular value (e.g., y = 1) can be expressed in terms of an explanatory (or regression) model through the link function, one of which is the logit, which provides log odds coefficients and corresponding odds ratios and is estimated through an iterative algorithm such as maximum likelihood (ML). Generalized linear models, therefore, avoid trying to transform the observed values of a binary or multinomial variable but, instead, apply a transformation to the expected values. Consider a case where we want to examine whether or not students are proficient in reading 1. We wish to see if there is a relationship between background variables (i.e., student gender, SES, race/ethnicity) and students likelihood to be proficient (coded 1 = proficient, 0 = not proficient). We might simply ask: Are students socioeconomic status, gender, and race/ethnicity related to their likelihood of being proficient in reading? We will begin with a simple discriminant analysis where we will try to separate students into two groups based on their gender and socioeconomic status. We probably will not have a great ability to classify since there are likely a number of other variables missing from our model. Discriminant Analysis Table 1. Wilks' Lambda Test of Function(s) Wilks' LambdaChi-square df Sig Download the data set (LogisticDichotomousData.zip) from the class web page. Instructions for replicating the corresponding models (tables 1-7) are provided at the end of this handout.

2 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories 2 We can see that the single function (since there are only two groups) is significant, which suggests that the set of three background variables is significantly related to classifying students into the two groups. The canonical correlation is 0.33 (not tabled). In the sample, there are 69.1% proficient and 30.9% who are not proficient (not tabled). If we examine the standardized coefficients we see that SES dominates in classifying students, followed by race ethnicity. Table 2. Standardized Canonical Discriminant Function Coefficients Function 1 lowses.706 female minor.506 We would likely classify almost 70% correctly by chance alone, so we can see that our simple model is really not doing any better than that. In particular, it is not very accurate in classifying individuals who are not proficient (with only 26% of the non-proficient students being correctly classified). Table 3. Classification Results b,c Predicted Group Membership readprof 0 1 Total Original Count % Cross-validated a Count % a. Cross validation is done only for those cases in the analysis. In cross validation, each case is classified by the functions derived from all cases other than that case. b. 70.5% of original grouped cases correctly classified. c. 70.5% of cross-validated grouped cases correctly classified.

3 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories 3 Next, we can use SPSS to fit a number of different models with various types of categorical outcomes (e.g., dichotomous, multinomial, count, ordinal). As Hox notes, generalized linear models include the necessary transformation and appropriate error distribution within the statistical model. They have three common components: An outcome variable Y with a specific error distribution with mean μ and 2 variance σ, A linear additive regression equation that produces a transformed predictor η of Y, and a link function which connects the expected values of Y to the predicted η = f ( μ) values of η :. Depending on the sampling distribution of the outcome variable, particular error distributions (e.g., normal, binomial or Bernoulli, Poisson) are incorporated into the particular link function chosen. In the case where the link function is identity and the errors are normally distributed (as when the outcome is continuous), no transformation of the outcome is needed. The generalized linear model simplifies to the familiar multiple regression model. For models where the outcome is categorical, this expected value can be transformed so the predictions are constrained to lie within a particular interval. Generalized linear models therefore make possible the extension of the linear model with a continuous outcome to situations where the outcome has some type of non-normal error distribution. In these cases, the appropriate selection of a non-normal error distribution and relevant nonlinear link function can provide more efficient estimates of model parameters. Dichotomous Outcome Where the outcome is dichotomous, the sampling model is binomial (Hox, 2002). One more appropriate choice is logistic regression (e.g., another possibility is probit regression, which uses ln( μ /(1 μ )) a probit link function). The link function is a logit function given by η =, and the transformed predicted value η of the outcome Y can be represented through a linear structural η = β + β X + β X model of the form Logistic regression is based on the probability that the dichotomous outcome Y is either 0 or 1. If the population proportion of cases for which Y = 1 is defined as the probabilityπ = 1, then the probability that Y = 0 can be defined as 1 π. This can be expressed as a linear logit equation, where in the binary case, the logit, or log odds, for the event is a linear expression which can be written in terms of a logit function. Taking the logarithm of the odds provides a way of representing the additive effects of the set of predictors on the outcome. The log odds (η i ) for the likelihood of individual i being proficient in reading can be written as follows: π Loge = β0 + β1( SES) i + β2( female) i + β3( minor) i 1 π. (1) η i = Equation 1 suggests a log odds coefficient is a ratio of two probabilities. The ratio

4 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories 4 π/(1 π) is defined as the odds for y = 1 as opposed to y = 0. For example, if the probability of being proficient is 0.5, then 0.5/0.5 = 1, and the corresponding log (1) = 0. This suggests each even is equally likely to occur. If the probability of being proficient is 0.9, then the odds will be greater than 1.0 (0.9/1-0.9 = 9), and log(9) = Conversely, if the probability is less than 0.5 the odds Y = 1 will be less than 1.0. Therefore, although the predicted value for the transformed outcome can take on any real value, the probability y = 1 will vary between 0 and 1. The usual residual variance (e i in a typical regression model) is not included in the logistic regression model represented in Eq. 1 because for a binomial distribution the residual variance is a function of the population mean (or proportion) and cannot be estimated separately (Hox, 2002). It is typically set to a scale factor of 1.0, which suggests it does not need to be interpreted (Hox, 2010). One of the desirable features of the logit link is that odds ratios can be obtained ( e β ), where e is approximately and β is the specific log odds coefficient (so if the log odds of β = 0, the odds ratio equals 1). Odds ratios are typically easier to interpret than log odds. The log odds can also be used to estimate the predicted probability Y = 1 from the log odds coefficient. 1 π = x x ( ) e β 0 β β + (2) where βs are logistic regression coefficients for the intercept and two covariates. This model assures that whether the predicted value for η is positive, negative, or zero, the resulting predicted probability will lie between 0 and 1. In the table following, where we just have the simple intercept log odds for the proficiency example (-0.803), we can estimate the predicted probability of being proficient as (where (0.803) =0.448) and 1/1.448 = Table 4. Parameter Estimates 95% Wald Confidence Interval Hypothesis Test Wald Chi- Parameter B Std. Error Lower Upper Square df Sig. (Intercept) (Scale) 1 a Dependent Variable: readprof Model: (Intercept) a. Fixed at the displayed value. The estimated probability of a student being proficient in math (0.691) matches the percentage proficient in the table below. Table 5. Categorical Variable Information N Percent Dependent Variable readprof % % Total %

5 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories 5 When we estimate this preliminary model, we obtain the following results. First, it is important to note that the defaults in IBM SPSS use the last category as the reference group. For example, in the model, this would be students with low SES background, females, minority by race/ethnicity, and proficient in reading (coded 1). We can, however, change the reference group to be the groups coded 0. This is consistent, for example, with dummy coding, where the reference category is coded 0 and the named category (e.g., female) is coded 1. We can then obtain estimates for the categories coded 1 (as shown in the table below). Table 6. Parameter Estimates 95% Wald 95% Wald Confidence Interval Hypothesis Test Confidence Interval for Exp(B) Parameter B Std. Error Lower Upper Wald Chi- Square df Sig. Exp(B) Lower Upper (Intercept) [lowses=1] [lowses=0] 0 a [female=1] [female=0] 0 a [minor=1] [minor=0] 0 a (Scale) 1 b Dependent Variable: readprof Model: (Intercept), lowses, female, minor a. Set to zero because this parameter is redundant. b. Fixed at the displayed value. We can interpret the intercept (1.545) as the predicted log odds of Y = 1 if all the predictors were equal to 0. That individual would be average/high SES (0), male (0), and not a minority by race/ethnicity (0). Regarding gender, since males as the reference group, the table suggests females are significantly more likely to be proficient in reading ( β = 0.465, p <.01) than males. The odds ratio (1.592) suggests that females have about a 59.2% increased likelihood to be proficient compared with males. Stated differently, the odds of being proficient are increased by about 1.6 times for females compared to males. For comparative purposes, if the odds ratio was 2:1, this would represent a 100% increase in the odds of being proficient compared to males. In contrast, low SES is negatively related to likelihood to be proficient ( β = , p <.01). Expressed as an odds ratio, the odds of a student who is low SES being proficient are reduced by a factor of (or 65.1%) compared to the reference group. Odds ratios below 1.0 are sometimes easier to explain by converting them into the odds of not being proficient. This can be accomplished by dividing 1 by the obtained odds ratio (1/.349). This will change the odds ratio to 2.865, representing the odds of being not proficient. This suggests the odds of being proficient

6 Ronald H. Heck 6 EDEP 606: Multivariate Methods (2013) April 23, 2013 for low SES students are reduced by almost 2.9 times compared with their peers of average or high SES background. In addition to examining the statistical significance of the individual variables, we can also obtain other information to evaluate how well the model fits the data., such as the ability to classify individuals correctly into likelihood to be proficient or not, or obtaining the model s log likelihood, which can be used to evaluate a series of model tests. Models with smaller log likelihoods represent better-fitting models. Using the overall percentage of individuals correctly classified, in the table below, we note this model also correctly classified 70.5% of the participants (again with greater accuracy for proficient students). Table 7. Classification Table a Predicted readprof Percentage Observed 0 1 Correct Step 1 readprof Overall Percentage 70.5 a. The cut value is.500 References Agresti, A. (2007). An introduction to categorical data analysis. Hoboken, NJ: John Wiley & Sons, Inc. Hox, J. (2010). Multilevel analysis: Techniques and applications (2nd ed.). NY: Routledge Academic. McCullagh, P., & Nelder, J. A. (1989). Generalized linear models (2nd ed.). New York: Chapman and Hall.

7 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories 7 Defining the Discriminant Analysis Model (Tables 1, 2, 3) with IBM SPSS Menu Commands IBM SPSS syntax: DISCRIMINANT /GROUPS=readprof(0 1) /VARIABLES=lowses female minor /ANALYSIS ALL /PRIORS SIZE /STATISTICS=TABLE CROSSVALID /CLASSIFY=NONMISSING POOLED. Launch the IBM SPSS application program and select the LogisticDichotomous.sav data file. 1. Go to the toolbar, select ANALYZE, CLASSIFY, DISCRIMINANT. This command opens the Discriminant Analysis main dialog box.

8 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories 8 2. In the Discriminant Analysis main dialog box we will specific the grouping variable and independent variables in the model. a. We will specify readprof as the grouping variable. Click to select readprof then click the right arrow button (or drag the variable) into the Grouping Variable box. b. Now we need to define the range of the grouping variable (readprof). Click DEFINE RANGE button to access the dialog box. c. In the Discriminant Analysis Define Range dialog box, enter the minimum and maximum values of readprof (0,1). Then click the CONTINUE button to close the dialog box. d. Now designate the predictors for the model. Click to select lowses, female, and minor then click the right arrow button (or drag the variables) into the Independents box. Note: The default is Enter independents together which we ll retain for this analysis. A stepwise analysis would require using the stepwise method instead. Note: Although not necessary for this example if you would like to include the means in the output, click the STATISTICS button to access the dialog box and select Means.

9 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories 9 3a. From the Discriminant Analysis main dialog box click the CLASSIFY button to access the dialog box. b. For this example change the default prior probabilities setting by selecting Compute from group means. This option takes the observed group size in the sample to determine the prior probabilities of group membership (IBM SPSS, 2011). c. For the display we will select Summary table which will be included in the output. d. We will also select Leave-one-out classification. This option denotes that each case in the analysis is classified by the functions derived from all cases other than that case (IBM SPSS, 2011). Click the CONTINUE button to return to the Discriminant Analysis main dialog box. 4. From the Discriminant Analysis main dialog box click the OK button to generate the output results.

10 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories 10 Defining Logistic Regression Analysis Model (Tables 4, 5) with IBM SPSS Menu Commands IBM SPSS syntax: GENLIN readprof (REFERENCE=FIRST) /MODEL INTERCEPT=YES DISTRIBUTION=BINOMIAL LINK=LOGIT /CRITERIA METHOD=FISHER(1) SCALE=1 COVB=MODEL MAXITERATIONS=100 MAXSTEPHALVING=5 PCONVERGE=1E-006(ABSOLUTE) SINGULAR=1E-012 ANALYSISTYPE=3(WALD) CILEVEL=95 CITYPE=WALD LIKELIHOOD=FULL /MISSING CLASSMISSING=EXCLUDE /PRINT CPS DESCRIPTIVES MODELINFO FIT SUMMARY SOLUTION (EXPONENTIATED). (Continue using the LogisticDichotomous.sav data file.) 1. Go to the toolbar and select ANALYZE, GENERALIZED LINEAR MODELS, GENERALIZED LINEAR MODELS. This command opens the Generalized Linear Models (GENLIN) main dialog box.

11 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories 11 2a. The Generalized Linear Models display screen displays nine command tabs: Type of Model, Response, Predictors, Model, Estimation, Statistics, EM Means, Save, Export. The default command tab when creating a model for the firsttime is Type of Model and enables specifying a model s distribution of the dependent variable and the link function b. For this first model we will designate a binary logistic analysis due to the dichotomous aspect of the dependent variable (readprof) to be used for this model. Click to select: Binary logistic. 3a. Click the Response command tab which enables specifying a dependent variable for the model. b. For this model we will use the reading proficiency scores of students as the dependent variable. Click to select readprof from the Variables list then click the right arrow button to move the variable into the Dependent Variable box. c. When the dependent variable has only two values we can specify a reference category for parameter estimation. To specify a reference category click the REFERENCE CATEGORY button to activate the Generalized Linear Models Reference Category dialog box to appear on screen.

12 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories 12 d. We will change the reference category by clicking to select: First (lowest value). Interpretation of the resulting parameter estimates will be relative to the likelihood of the value 0 category. e. Click the CONTINUE button to close the Generalized Linear Models Reference Category dialog box Since this is a no-predictors (null) model we will skip-over the Predictors and Model command tabs. We will also retain the default settings in the Estimation command tab which enables specifying estimation methods and providing initial values for the parameter estimates. 4a. Click the Statistics command tab which enables specifying model effects (analysis type and confidence intervals) and print output. b. We want to have odds ratios included in the output so click to select Include exponential parameter estimates.. c. Click the OK button to generate the model results. A warning message is displaced to remind the user that the model has no effects. Click YES to generate the model output results.

13 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories 13 Defining the Logistic Regression Analysis Model (Table 7) with IBM SPSS Menu Commands IBM SPSS syntax: LOGISTIC REGRESSION VARIABLES readprof /METHOD=ENTER lowses female minor /CONTRAST (lowses)=indicator /CONTRAST (female)=indicator /CONTRAST (minor)=indicator /CRITERIA=PIN(.05) POUT(.10) ITERATE(20) CUT(.5). (Continue using the LogisticDichotomous.sav data file.) 1. Go to the toolbar, select REGRESSION, BINARY LOGISTIC. This command opens the Logistic Regression main dialog box.

14 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories In the Logistic Regression main dialog box we will specific the dependent variable and independent variables in the model. a. We will specify readprof as the dependent variable. Click to select readprof then click the right arrow button (or drag the variable) into the Dependent box. b. Next we need to add the predictor variables to the model. Click to select lowses, female, and minor then click the right arrow button to add them to the Covariates box c. Since lowses, female, and minor are coded 0, 1, we will specify them as categorical variables. Click the CATEGORICAL button to access the Logistic Regression Define Categorical Variables. d. Click to select lowses, female, and minor then click the right arrow button to add them to the Categorical Covariates box. e. Adding the variables into the Categorical Covariates box now activates the Change Contrast options. We will retain the default setting. Click the CONTINUE button to return to the Logistic Regression main dialog box.

15 Ronald H. Heck and Lynn N. Tabata Investigating Models with Two or Three Categories Click the OK button to generate the output results.

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