The SPSS Sample Problem To demonstrate these concepts, we will work the sample problem for logistic regression in SPSS Professional Statistics 7.5, pa

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1 The SPSS Sample Problem To demonstrate these concepts, we will work the sample problem for logistic regression in SPSS Professional Statistics 7.5, pages The description of the problem can be found on page 39. The data for this problem is: Prostate.Sav. Stage One: Define the Research Problem In this stage, the following issues are addressed: Relationship to be analyzed Specifying 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). 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 reached advanced stage' GRADE 'Aggressive tumor' Method for including independent variables 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. Stage 2: Develop the Analysis Plan: Sample Size Issues In this stage, the following issues are addressed: Missing data analysis Minimum sample size requirement: cases per independent variable Missing data analysis There is no missing data in this problem.

2 Minimum sample size requirement: 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 cases per independent variable. We should look for opportunities to validate our findings against other samples before generalizing our results. Stage 2: Develop the Analysis Plan: Measurement Issues: In this stage, the following issues are addressed: Incorporating nonmetric data with dummy variables Representing Curvilinear Effects with Polynomials Representing Interaction or Moderator 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. Stage 3: Evaluate Underlying Assumptions In this stage, the following issues are addressed: Nonmetric dependent variable with two groups Metric or dummy-coded independent variables 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. 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.

3 Requesting a Logistic Regression First, choose the Regression > Binary Logistic...' command from the 'Analyze' menu.

4 Specifying the Dependent Variable First, move the dependent variable NODALINV 'Cancer spread to lymph nodes' to the 'Dependent:' text box. Specifying the Independent Variables First, move the independent variables to the Covariates:' list box. AGE 'Age of the subject' ACID 'Laboratory test score' XRAY 'Postive Xray result' STAGE 'Disease reached advanced stage' GRADE 'Aggressive tumor'

5 Specify the method for entering variables First, accept the 'Enter' default entry method from the 'Method:' popup menu.

6 Specifying Options to Include in the Output Second, in the 'Statistics and Plots' panel, mark the 'Classificatio n plots' checkbox, the 'Hosmer- Lemeshow goodnessof-fit' check box, and the 'Casewise listing of residuals' option button with the 'Outliers outside 2 std. dev. option. Third, mark the option button 'At each step' on the 'Display' panel to display output every time a variable is added. First, click on the 'Options...' button to open the Logistic Regression: Options dialog box, Fifth, accept the other defaults and click on the Continue button. Fourth, check the Iteration history to obtain the starting model.

7 Specifying the New Variables to Save First, click on the 'Save...' button to open the 'Logistic Regression: Save New Variables' dialog box. Third, click on the 'Continue' button. Complete the Logistic Regression Request Second, mark the 'Cook's' checkbox on the 'Influence' panel. First, click on the OK button to complete the logistic regression request. 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 likelihood (Change in -2LL) Measures Analogous to R 2: Cox and Snell R 2 and Nagelkerke R 2

8 Hosmer-Lemeshow Goodness-of-fit Classification matrices as a measure of model accuracy Check for Numerical Problems Presence of outliers 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 on step 0, before any variables have been added to the model. Significance test of the model log likelihood The difference between these two measures is the model child-square value ( = ) that is tested for statistical significance. This test is analogous to the F-test for R 2 or change in R 2 value in multiple regression which tests whether or not the improvement in the model associated with the additional variables is statistically significant.

9 In this problem the model Chi-Square value of 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. Measures Analogous to R 2 The next SPSS outputs indicate the strength of the relationship between the dependent variable and the independent variables, analogous to the R 2 measures in multiple regression. The Cox and Snell R 2 measure operates like R 2, 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 interpretive criteria to the Nagelkerke R 2 of 0.465, we would characterize the relationship as strong. 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 which has the desirable outcome of nonsignificance. 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.

10 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 compute the proportional by chance accuracy rate and the maximum by chance accuracy rates, if appropriate. The proportional by chance accuracy rate is equal to (0.623^ ^2). A 25% increase over the proportional by chance accuracy rate would equal Our model accuracy race of 77.4% meets this criterion. Since one of our groups contains 62.3% of the cases, we might also apply the maximum by chance criterion. A 25% increase over the largest groups would equal Our model accuracy race of 77.4% almost meets this criterion. SPSS provides a visual image of the classification accuracy in the stacked histogram as shown below. To the extent to which the cases in one group cluster on the left and the other group clusters on the right, the predictive accuracy of the model will be higher.

11 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 dummycoded 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.

12 The standard errors and B coefficients are not excessively large, so there is no evidence of a numeric problem with this analysis. 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:

13 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. Request the Scatterplot First, select the 'Scatter...' command from the Graphs menu. Second, highlight the thumbnail sketch of the 'Simple' scatterplot. Third, click on the Define button to specify the variables for the scatterplot.

14 Specifying the Variables for the Scatterplot First, move the variable COO_1 to the text box for the 'Y Axis:' variable. Second, move the case variable to the 'X Axis' text box. Third, click on the OK button to complete the request 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.

15 Stage 5: Interpret the Results In this section, we address the following issues: In this section, we address the following issues: Identifying the statistically significant predictor variables Direction of relationship and contribution to dependent variable 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.

16 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. 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. 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 Validation Analysis Computing the Second Validation Analysis 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.

17 Set the Starting Point for Random Number Generation Second, click on the 'Set seed to:' option to access the text box for the seed number. First, select the 'Random Number Seed...' command from the 'Transform' menu. Fourth, click on the OK button to complete this action. Third, accept the default ' ' in the 'Set seed to:' text box.

18 Compute the Variable to Select a Large Proportion of the Data Set First, select the 'Compute...' command from the Transform menu. Second, create a new variable named 'split1 that has the values 1 and 0 to divide the sample into two parts. Type the name 'split1' into the 'Target Variable:' text box. Third, type the formula 'uniform(1) > 0.15' in the 'Numeric Expression:' text box. The uniform function will generate a random number between 0.0 and 1.0 for each case. If the generated random number is greater than 0.15, the numeric expression will result in a 1, since the numeric expression is true. We will include cases with a split value of 1 in the validation analysis. Fourth, we click on the OK button to compute the split variable.

19 Specify the Cases to Include in the First Validation Analysis First, select 'Logistic Regression' from the 'Dialog Recall' drop down menu.

20 Specify the Value of the Selection Variable for the First Validation Analysis First, click on the 'Select>>" button to expose the 'Selection Variable:' text box. Fourth, type a '1' in the 'Value for Selection Variable:' text box. Second, highlight the 'split1' variable and click on the move button to put it into the 'Selection Variable:' text box. Only Fifth, click on the 'Continue' button to complete setting the value. Third, after 'split1=?' appears in the 'Selection Variable:' text box, click on the Value..' button to specify which cases to include in the screening sample. 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% sampleḋraft of the

21 Set the Starting Point for Random Number Generation Second, click on the 'Set seed to:' option to access the text box for the seed number. First, select the 'Random Number Seed...' command from the 'Transform' menu. Fourth, click on the OK button to complete this action. Third, type ' ' in the 'Set seed to:' text box.

22 Compute the Variable to Select a Large Proportion of the Data Set First, select the 'Compute...' command from the Transform menu. Second, create a new variable named 'split2' that has the values 1 and 0 to divide the sample into two parts. Type the name 'split2' into the 'Target Variable:' text box. Fourth, we click on the OK button to compute the split variable. Third, type the formula 'uniform(1) > 0.15' in the 'Numeric Expression:' text box. The uniform function will generate a random number between 0.0 and 1.0 for each case. If the generated random number is greater than 0.15, the numeric expression will result in a 1, since the numeric expression is true. We will include cases with a split value of 1 in the validation analysis.

23 Specify the Cases to Include in the Second Validation Analysis First, select 'Logistic Regression' from the 'Dialog Recall' drop down menu.

24 Specify the Value of the Selection Variable for the Second Validation Analysis First, highlight the 'split1=1' text in the 'Selection Variable:' text box and click on the move button to return the variable to the list of variables. Second, highlight the 'split2' variable and click on the move button to put it into the 'Selection Variable:' text box. Generalizability of the Logistic Regression Model Fourth, type a '1' in the 'Value for Selection Variable:' text box. Third, after 'split2=?' appears in the 'Selection Variable:' text box, click on the Value..' button to specify which cases to include in the screening sample. We can summarize the results of the validation analyses in the following table. Fifth, click on the 'Continue' button to complete setting the value. Full Model Split1 = 1 Split2 = 1 Model Chi-Square , p= , p= , p=.0053 Nagelkerke R Accuracy Rate for Learning Sample 77.36% 75.00% 75.56% Accuracy Rate for Validation Sample 76.92% 87.50% Significant STAGE 'Disease STAGE 'Disease STAGE 'Disease Coefficients (p < 0.05) reached advanced stage' and the reached advanced stage' and the reached advanced stage' and the variable XRAY 'Positive Xray result' variable (0.0531) XRAY 'Positive Xray result' variable XRAY 'Positive Xray result'

25 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 analyses, I would conclude that our results are generalizable.