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The SPSS Sample Problem

To demonstrate these concepts, we will work the

sample problem for logistic regression in SPSS

Professional Statistics 7.5, pages 37 - 64. The

description of the problem can be found on page

39. The data for this problem is Prostate.Sav.

SPSS Sample Problem

Stage One Define the Research Problem

In this stage, the following issues are

addressed Relationship to be analyzed Specify

ing the dependent and independent variables

Method for including independent variables

Relationship to be analyzed

The goal of this analysis is to determine the

relationship between the dependent variable

NODALINV (whether or not the cancer has spread to

the lymph nodes), and the independent variables

of AGE (age of the subject), ACID (a laboratory

test value that is elevated when the tumor has

spread to certain areas), STAGE (whether or not

the disease has reached an advanced stage), GRADE

(aggressiveness of the tumor), and XRAY (positive

or negative xray result).

SPSS Sample Problem

Method for including independent variables

Specifying the dependent and independent variables

The dependent variable is NODALINV 'Cancer

spread to lymph nodes', a dichotomous variable.

The independent variables are AGE 'Age of

the subject' ACID 'Laboratory test score' XRAY

'Positive X-ray result' STAGE 'Disease reach

ed advanced stage' GRADE 'Aggressive tumor'

Since we are interested in the relationship

between the dependent variable and all of the

independent variables, we will use direct entry

of the independent variables.

SPSS Sample Problem

Stage 2 Develop the Analysis Plan Sample Size

Issues

In this stage, the following issues are

addressed Missing data analysis Minimum sampl

e size requirement 15-20 cases per independent

variable

Missing data analysis

There is no missing data in this problem.

Minimum sample size requirement15-20 cases per

independent variable

The data set has 53 cases and 5 independent

variables for a ratio of 10 to 1, short of the

requirement that we have 15-20 cases per

independent variable. We should look for

opportunities to validate our findings against

other samples before generalizing our results.

SPSS Sample Problem

Stage 2 Develop the Analysis Plan Measurement

Issues

In this stage, the following issues are

addressed Incorporating nonmetric data with du

mmy variables Representing Curvilinear Effects wi

th Polynomials Representing Interaction or Modera

tor Effects

Incorporating Nonmetric Data with Dummy Variables

All of the nonmetric variables have recoded into

dichotomous dummy-coded variables.

Representing Curvilinear Effects with Polynomials

We do not have any evidence of curvilinear

effects at this point in the analysis.

Representing Interaction or Moderator Effects

We do not have any evidence at this point in the

analysis that we should add interaction or

moderator variables.

SPSS Sample Problem

Stage 3 Evaluate Underlying Assumptions

In this stage, the following issues are

addressed Nonmetric dependent variable with tw

o groups Metric or dummy-coded independent variab

les

Nonmetric dependent variable having two groups

The dependent variable NODALINV 'Cancer spread

to lymph nodes' is a dichotomous variable.

Metric or dummy-coded independent variables

AGE 'Age of the subject' and ACID 'Laboratory

test score' are metric variables.

XRAY 'Positive X-ray result', STAGE 'Disease

reached advanced stage', and GRADE 'Aggressive

tumor' are nonmetric dichotomous variables.

SPSS Sample Problem

Stage 4 Estimation of Logistic Regression and

Assessing Overall Fit Model Estimation

In this stage, the following issues are

addressed

Compute logistic regression model

Compute the logistic regression

The steps to obtain a logistic regression

analysis are detailed on the following screens.

SPSS Sample Problem

Requesting a Logistic Regression

SPSS Sample Problem

Specifying the Dependent Variable

SPSS Sample Problem

Specifying the Independent Variables

SPSS Sample Problem

Specify the method for entering variables

SPSS Sample Problem

Specifying Options to Include in the Output

SPSS Sample Problem

Specifying the New Variables to Save

SPSS Sample Problem

Complete the Logistic Regression Request

SPSS Sample Problem

Stage 4 Estimation of Logistic Regression and

Assessing Overall Fit Assessing Model Fit

In this stage, the following issues are

addressed Significance test of the model log l

ikelihood (Change in -2LL) Measures Analogous to

R² Cox and Snell R² and Nagelkerke R²

Hosmer-Lemeshow Goodness-of-fit

Classification matrices as a measure of model

accuracy Check for Numerical Problems Presence o

f outliers

SPSS Sample Problem

Initial statistics before independent variables

are included

The Initial Log Likelihood Function, (-2 Log

Likelihood or -2LL) is a statistical measure like

total sums of squares in regression. If our

independent variables have a relationship to the

dependent variable, we will improve our ability

to predict the dependent variable accurately, and

the log likelihood value will decrease. The

initial 2LL value is 70.252 on step 0, before

any variables have been added to the model.

SPSS Sample Problem

Significance test of the model log likelihood

The difference between these two measures is the

model child-square value (22.126 70.252 -

48.126) that is tested for statistical

significance. This test is analogous to the

F-test for R² or change in R² value in multiple

regression which tests whether or not the

improvement in the model associated with the

additional variables is statistically significant.

In this problem the model Chi-Square value of

22.126 has a significance of 0.000, less than

0.05, so we conclude that there is a significant

relationship between the dependent variable and

the set of independent variables.

SPSS Sample Problem

Measures Analogous to R²

The next SPSS outputs indicate the strength of

the relationship between the dependent variable

and the independent variables, analogous to the

R² measures in multiple regression.

The Cox and Snell R² measure operates like R²,

with higher values indicating greater model fit.

However, this measure is limited in that it

cannot reach the maximum value of 1, so

Nagelkerke proposed a modification that had the

range from 0 to 1. We will rely upon

Nagelkerke's measure as indicating the strength

of the relationship. If we applied our interpre

tive criteria to the Nagelkerke R² of 0.465, we

would characterize the relationship as strong.

SPSS Sample Problem

Correspondence of Actual and Predicted Values of

the Dependent Variable

The final measure of model fit is the Hosmer and

Lemeshow goodness-of-fit statistic, which

measures the correspondence between the actual

and predicted values of the dependent variable.

In this case, better model fit is indicated by a

smaller difference in the observed and predicted

classification. A good model fit is indicated by

a nonsignificant chi-square value.

The goodness-of-fit measure has a value of 5.954

which has the desirable outcome of

nonsignificance.

SPSS Sample Problem

The Classification Matrices as a Measure of Model

Accuracy

The classification matrices in logistic

regression serve the same function as the

classification matrices in discriminant analysis,

i.e. evaluating the accuracy of the model.

If the predicted and actual group memberships are

the same, i.e. 1 and 1 or 0 and 0, then the

prediction is accurate for that case. If

predicted group membership and actual group

membership are different, the model "misses" for

that case. The overall percentage of accurate

predictions (77.4 in this case) is the measure

of a model that I rely on most heavily for this

analysis as well as for discriminant analysis

because it has a meaning that is readily

communicated, i.e. the percentage of cases for

which our model predicts accurately.

To evaluate the accuracy of the model, we comput

e the proportional by chance accuracy rate and

the maximum by chance accuracy rates, if

appropriate. The proportional by chance accur

acy rate is equal to 0.530 (0.6232 0.3772).

A 25 increase over the proportional by chance

accuracy rate would equal 0.663. Our model

accuracy race of 77.4 meets this criterion.

Since one of our groups contains 62.3 of the ca

ses, we might also apply the maximum by chance

criterion. A 25 increase over the largest

groups would equal 0.778. Our model accuracy

race of 77.4 almost meets this criterion.

Stacked Histogram

SPSS provides a visual image of the

classification accuracy in the stacked histogram

as shown below. To the extent to which the ca

ses in one group cluster on the left and the

other group clusters on the right, the predictive

accuracy of the model will be higher.

Check for Numerical Problems

There are several numerical problems that can

occur in logistic regression that are not

detected by SPSS or other statistical packages

multicollinearity among the independent

variables, zero cells for a dummy-coded

independent variable because all of the subjects

have the same value for the variable, and

"complete separation" whereby the two groups in

the dependent event variable can be perfectly

separated by scores on one of the independent

variables. All of these problems produce large

standard errors (over 2) for the variables

included in the analysis and very often produce

very large B coefficients as well. If we

encounter large standard errors for the predictor

variables, we should examine frequency tables,

one-way ANOVAs, and correlations for the

variables involved to try to identify the source

of the problem.

The standard errors and B coefficients are not

excessively large, so there is no evidence of a

numeric problem with this analysis.

SPSS Sample Problem

Presence of outliers

There are two outputs to alert us to outliers

that we might consider excluding from the

analysis listing of residuals and saving Cook's

distance scores to the data set.

SPSS provides a casewise list of residuals that

identify cases whose residual is above or below a

certain number of standard deviation units. Like

multiple regression there are a variety of ways

to compute the residual. In logistic regression,

the residual is the difference between the

observed probability of the dependent variable

event and the predicted probability based on the

model. The standardized residual is the residual

divided by an estimate of its standard

deviation. The deviance is calculated by taking

the square root of -2 x the log of the predicted

probability for the observed group and attaching

a negative sign if the event did not occur for

that case. Large values for deviance indicate

that the model does not fit the case well. The

studentized residual for a case is the change in

the model deviance if the case is excluded.

Discrepancies between the deviance and the

studentized residual may identify unusual cases.

(See the SPSS chapter on Logistic Regression

Analysis for additional details, pages 57-61).

In the output for our problem, SPSS listed three

cases that have may be considered outliers with

a studentized residuals greater than 2

SPSS Sample Problem

Cooks Distance

SPSS has an option to compute Cook's distance as

a measure of influential cases and add the score

to the data editor. I am not aware of a precise

formula for determining what cutoff value should

be used, so we will rely on the more traditional

method for interpreting Cook's distance which is

to identify cases that either have a score of 1.0

or higher, or cases which have a Cook's distance

substantially different from the other. The

prescribed method for detecting unusually large

Cook's distance scores is to create a scatterplot

of Cook's distance scores versus case id.

SPSS Sample Problem

Request the Scatterplot

SPSS Sample Problem

Specifying the Variables for the Scatterplot

SPSS Sample Problem

The Scatterplot of Cook's Distances

On the plot of Cook's distances, we see a case

that exceeds the 1.0 rule of thumb for

influential cases. Scanning the data in the data

editor, we find that the case with the large

Cook's distance is case 24. If we study case 24

in the data editor, we will find that this case

had the highest score for the acid variable, but

no nodal involvement. Comparing this case to the

two cases with the next highest acid score, case

25 with a score of 136 and case 53 with a score

of 126, we see that both of these cases had nodal

involvement, suggesting that a high acid score is

associated with nodal involvement. We can

consider case 24 as a candidate for exclusion

from the analysis.

SPSS Sample Problem

Stage 5 Interpret the Results

In this section, we address the following

issues Identifying the statistically significa

nt predictor variables Direction of relationship

and contribution to dependent variable

SPSS Sample Problem

Identifying the statistically significant

predictor variables

The coefficients are found in the column labeled

B, and the test that the coefficient is not zero,

i.e. changes the odds of the dependent variable

event is tested with the Wald statistic, instead

of the t-test as was done for the individual B

coefficients in the multiple regression equation.

Similar to the output for a regression equation,

we examine the probabilities of the test

statistic in the column labeled "Sig," where we

identity that the variable STAGE 'Disease

reached advanced stage' and the variable XRAY

'Positive X-ray result' have a statistically

significant relationship with the dependent

variable.

SPSS Sample Problem

Direction of relationship and contribution to

dependent variable

The signs of both of the statistically

significant independent variables are positive,

indicating a direct relationship with the

dependent variable. Our interpretation of these

variables is that positive (yes or 1) values to

both questions XRAY 'Positive Xray result' and

STAGE 'Disease reached advanced stage' are

associated with the positive (yes or 1) category

of the dependent variable NODALINV 'Cancer

spread to lymph nodes'.

Interpretation of the independent variables is

aided by the "Exp (B)" column which contains the

odds ratio for each independent variable. Thus,

we would say that persons with a value of 1

STAGE 'Disease reached advanced stage' are 4.77

times as likely to have a score of 1 on the

dependent variable NODALINV 'Cancer spread to

lymph nodes'. Similarly, persons whose score is 1

on the independent variable XRAY 'Positive

X-ray result' have a 7.73 greater likelihood of

having lymph node involvement.

SPSS Sample Problem

Stage 6 Validate The Model

When we have a small sample in the full data set

as we do in this problem, a split half validation

analysis is almost guaranteed to fail because we

will have little power to detect statistical

differences in analyses of the validation

samples. In this circumstance, our alternative

is to conduct validation analyses with random

samples that comprise the majority of the

sample. We will demonstrate this procedure in

the following steps Computing the First Valid

ation Analysis Computing the Second Validation An

alysis The Output for the Validation Analysis

Computing the First Validation Analysis

We set the random number seed and modify our

selection variable so that is selects about

75-80 of the sample.

SPSS Sample Problem

Set the Starting Point for Random Number

Generation

SPSS Sample Problem

Compute the Variable to Select a Large Proportion

of the Data Set

SPSS Sample Problem

Specify the Cases to Include in the First

Validation Analysis

SPSS Sample Problem

Specify the Value of the Selection Variable for

the First Validation Analysis

SPSS Sample Problem

Computing the Second Validation Analysis

We reset the random number seed to another value

and modify our selection variable so that is

selects about 75-80 of the sample.

SPSS Sample Problem

Set the Starting Point for Random Number

Generation

SPSS Sample Problem

Compute the Variable to Select a Large Proportion

of the Data Set

SPSS Sample Problem

Specify the Cases to Include in the Second

Validation Analysis

SPSS Sample Problem

Specify the Value of the Selection Variable for

the Second Validation Analysis

SPSS Sample Problem

Generalizability of the Logistic Regression Model

We can summarize the results of the validation

analyses in the following table.

As we can see in the table, the results for each

analysis are approximately the same, except that

the variable STAGE 'Disease reached advanced

stage' was not quite significant in the first

validation analysis. Based on the validation an

alyses, I would conclude that our results are

generalizable.

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