
PARTIAL LEAST SQUARES MODELING
Overview
Probit regression is method of working with categorical dependent variables whose underlying distribution is assumed to be normal. That is, the assumptions of probit regression are consistent with having a dichotomous dependent variable whose distribution is assumed to be a proxy for a true underlying continuous normal distribution. Probit regression has been extended to cover multinomial dependent variables (more than two nominal categories) and to cover ordinal categorical dependent variables. These extensions are sometimes labeled mlogit and ologit respectively. Probit regression is an umbrella term meaning different things in different contexts, though the common denominator is treating categorical dependent variables assumed to have an underlying normal distribution. In SPSS the following modules implement probit regression.
Ordinal probit regression
In SPSS, this is the Analyze > Generalized Linear Models > Generalized Linear Models menu selection. The generalized linear models (GZLM) module implements regression with any of many types of link function, including probit. Ordinal probit regression is implemented using a multinomial (ordinal) distribution with a cumulative probit link function. Response models, discussed below, may be implemented in GZLM. Ordinal probit regression is discussed below.
Probit signalresponse models
In SPSS, this is the Analyze > Regression > Ordinal menu selection. Signalresponse models involve subjects who receive a cue (a signal), which is either absent or present (coded 0 or 1). Subjects are then measured on subsequent responses, which are ordered categorical variables, which are the dependent variables for which the cue is the independent. For instance, in one signalresponse study, racial stereotypes were tested, where the signal (0, 1) was being told or not told beforehand that a subject in a scenario is AfricanAmerican. The responses were answers to items like whether the respondent later recalled a given topic (ex., "Rob has a brother who is a gang member"; "Rob is athletic"; etc.), where the answers ranged from 1 = "I am fairly positive that it was not in the paragraph" to 6 = "I am fairly positive it was in the paragraph". The signalresponse model tested whether the cue influenced item recall, evidencing a bias toward racial stereotypes. This SPSS module is a type of ordinal regression and is discussed below.
Probit response models
In SPSS, this is the Analyze > Regression > Probit menu selection. Probit response models are a specialized form on analysis for grouped data. The classic example is in medical research, where patients are grouped by dosage of medicine, and the desired response is recovery. To generalize, probit response models are ones in which there is a binary response, such as 1=response, 0=no response. These binary responses are aggregated in grouped data, yielding a count of the response=1 observations. This count is divided by total sample size for each group, giving proportions across groups (ex., proportion who recover in each dosage group). In this way the dependent is a count made into a percentage. The central objective of research is to see what the effect of one or more independent variables (covariates) is on these proportions, such as the effect of dosage on recovery. One of the main objectives of a probit response model is to determine what level of stimulus is needed to elicit a given proportion of response (ex., 50% recovery). Optionally there may be a grouping factor, such as gender or medication brand, so the researcher can compare the relative potency of the stimulus across groups. Response models are primarily used in experimental designs, where it is easier to form the needed groups. When used in quasiexperimental settings, it is necessary to add additional covariates as explicit control variables. This type of probit regression model is treated below.
Multilevel probit regression
In SPSS, this is the Analyze > Mixed Models> Generalized Linear menu choice.
The generalized linear mixed model (GLMM)model is similar to GZLM but incorporates the capacity for hierarchical probit modeling in which the fixed effects model at level 1 (ex., student level) is influenced by a level 2 (ex., school) grouping variable and other level 2 predictors. Multilevel probit models are discussed below.
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Below is the unformatted table of contents.
PROBIT REGRESSION AND RESPONSE MODELS Table of Contents Introduction 7 Overview 7 Ordinal probit regression 7 Probit signalresponse models 7 Probit response models 8 Multilevel probit regression 8 Key concepts and terms 9 Probit transformations 9 The cumulative normal distribution 9 Probit coefficients 10 Elasticity 10 Significance testing 11 Frequently asked questions 11 What about probit in Stata? 11 Binary and ordinal probit regression 13 Binary and ordinal probit regression models 13 Binary probit regression in generalized linear models 13 Example 13 Overview 13 Binary probit regression output in SPSS GZLM 22 Ordinal probit regression in generalized linear models 28 Overview 28 Example 28 SPSS setup 28 SPSS ordinal probit output 30 Ordinal regression with a probit link 33 Overview 33 SPSS setup 33 Output for ordinal regression with a probit link 36 Model fitting information, goodnessoffit, and pseudo Rsquare tables 36 Test of parallel lines 37 Parameter estimates table 38 Probit signalresponse models 39 Overview 39 Type of model 40 Equal variance vs. unequal variance signalresponse models 41 The detection parameter, d 44 Model fit 45 Locationscale models 47 Unequal variances model in SPSS 48 Probit Response Models 49 Overview 49 Key concepts and terms 50 Data setup 51 Models 52 Variables 53 Unit of analysis 53 Response frequency variable 54 Total observations variable 54 Factor 54 Covariate(s) 55 Weighting variable 55 Example 56 Example summary 56 Options 56 Outputs: Pearson goodnessoffit chisquare 58 Outputs: Parallelism test 59 Outputs: Transformed response plots 59 Outputs: Parameter estimates 60 Outputs: Natural response rate 61 Outputs: Cell counts and residuals 62 Outputs: Confidence limits 62 Outputs: Relative median potency (RMP) 64 Assumptions for probit response models 65 Variance in the response variable 65 Parallelism. 65 Linearity in the probit 66 Normal distribution 66 Stimulusresponse. 66 Conditional potencies 66 Independent observations 67 Adequate number of groups 67 No negative counts 67 Total >= response 67 Frequently asked questions for probit response models 68 What is the data setup for a probit response model? 68 What happens if I enter individual rather than grouped data into the Probit procedure in SPSS? 68 What is the SPSS syntax for the probit response model? 69 Couldn't we use OLS regression to create a response model? 69 Couldn't we use a ttest instead of probit? 69 Multilevel probit regression 70 Overview 70 Example 70 Sample size in GLMM 70 SPSS multilevel probit setup 71 Defining the subject structure of the data 71 The "Fields & Effects" tab 72 The "Build Options" tab 75 The "Model Options" tab 76 SPSS multilevel probit output 77 Model viewer 77 The "Model Summary" table 79 The "Data Structure" table 80 Predicted by Observed" plot 80 The "Classification" table 80 The "Fixed Effects" table and diagram 81 The "Fixed Coefficients" table and diagram 83 The "Random Effect Covariances" table 85 The "Estimated Means" table 88 Bibliography 90 Pagecount: 92