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Garson, G. D. (2012). Probit Regression and Response Models. Asheboro, NC: Statistical Associates Publishers.
 

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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 signal-response models

In SPSS, this is the Analyze > Regression > Ordinal menu selection. Signal-response 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 signal-response study, racial stereotypes were tested, where the signal (0, 1) was being told or not told beforehand that a subject in a scenario is African-American. 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 signal-response 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 quasi-experimental 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 signal-response 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 set-up	28
SPSS ordinal probit output	30
Ordinal regression with a probit link	33
Overview	33
SPSS set-up	33
Output for ordinal regression with a probit link	36
Model fitting information, goodness-of-fit, and pseudo R-square tables	36
Test of parallel lines	37
Parameter estimates table	38
Probit signal-response models	39
Overview	39
Type of model	40
Equal variance vs. unequal variance signal-response models	41
The detection parameter, d	44
Model fit	45
Location-scale 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 goodness-of-fit chi-square	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
Stimulus-response.	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 set-up 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 t-test instead of probit?	69
Multilevel probit regression	70
Overview	70
Example	70
Sample size in GLMM	70
SPSS multilevel probit set-up	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