CLA 2 Paper & PPT - Advanced Statistical Concepts and Business Analytics

Running Head: LOGISTIC REGRESSION AND DISCRIMINANT ANALYSIS 1

LOGISTIC REGRESSION AND DISCRIMINANT ANALYSIS 12

LOGISTIC REGRESSION AND DISCRIMINANT ANALYSIS

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Part I

Logistic Regression

The age and gender of guests in a nursing home were examined whether they are the cause of deaths in 2015. Data was collected for gender, age and whether the guest died or not. In this case, death is our dependent variable while age and gender are the independent variables.

Since the dependent variable “died” was categorical with two levels, this was an indication that logistic regression analysis was suitable for prediction in this study (Austin & Merlo, 2017). The assumption of a dichotomous dependent variable was met in this case where “died” took to values 0) No, 1) Yes. The assumption of one or more predictor variables was met. Age was quantitative reporting respective ages of the guests. Gender was categorical with two levels 0) Females and 1) Males.

Analysis

The collected data was analyzed using SPSS to examine the relationship between the predictor variables and the binary dependent variable. A sample of 284 guests was used for this study for easy analysis and generalizations. Table 1 shows the total number of participants and the valid sample that was utilized in this study.

Table 1: Case processing summary

Case Processing Summary
Unweighted Casesa	N	Percent
Selected Cases	Included in Analysis	284	100.0
	Missing Cases	0	.0
	Total	284	100.0
Unselected Cases	0	.0
Total	284	100.0
a. If weight is in effect, see classification table for the total number of cases.

Table 1 shows that there were no missing values and all the 284 guests were examined.

Table 2 shows the coding of the categorical independent variable. According to the results, females were more (175) compared to males (109).

Table 2: Coding of the independent variable.

Categorical Variables Codings
	Frequency	Parameter coding
		(1)
Gender	Female	175	1.000
	Male	109	.000

Table 3 examines whether the overall model was statistically significant.

Table 3: Omnibus test

Omnibus Tests of Model Coefficients
	Chi-square	df	Sig.
Step 1	Step	82.462	2	.000
	Block	82.462	2	.000
	Model	82.462	2	.000

According to the results, the overall model was statistically significant with χ2 (2) = 82.46, p < 0.001 (Warner, 2020). This implies that we can carry on with the analysis.

Explained variations was reported as shown in the model summary table 4.

Table 4: Model summary

Model Summary
Step	-2 Log likelihood	Cox & Snell R Square	Nagelkerke R Square
1	311.189a	.252	.336
a. Estimation terminated at iteration number 5 because parameter estimates changed by less than .001.

According to the analysis the Cox & Snell R squared and Nagelkerke R squared acts as linear regression pseudo R2. In this case, Nagelkerke R2 showed that 33.60% of variation was explained by the independent variables (Riley et al., 2021).

Hosmer and Lemeshow tests were determined to show whether the model fits the given data (Fagerland & Hosmer, 2017). Table 5 shows the result of the test. According to the results the model significantly fits the data since the significant value is greater than 0.05.

Table 5: Hosmer and Lemeshow test

Hosmer and Lemeshow Test
Step	Chi-square	df	Sig.
1	8.847	8	.355

Table 6 shows the classification of the variables correctly and incorrectly predicted by the model.

Table 6: Classification table

Classification Tablea
	Observed	Predicted
		Died	Percentage Correct
		No	Yes
Step 1	Died	No	103	37	73.6
		Yes	44	100	69.4
	Overall Percentage			71.5
a. The cut value is .500

According to the table 103 cases are observed as no deaths and correctly predicted as no deaths. 100 cases were observed as died and correctly predicted as deaths. However, 37 cases observed as no deaths were predicted as deaths. 44 observed deaths were predicted as no deaths. In overall, 71.50% of cases are correctly predicted by the model.

Table 6 shows the coefficients of the variables in the model.

Table 6: Coefficients

Variables in the Equation
	B	S.E.	Wald	df	Sig.	Exp(B)
Step 1a	Gender(1)	-1.957	.323	36.596	1	.000	.141
	Age	.196	.030	43.740	1	.000	1.216
	Constant	-13.099	2.063	40.318	1	.000	.000
a. Variable(s) entered on step 1: Gender, Age.

Analysis showed that gender and age were both statistically significant and contributed to the variation in deaths. Gender was statistically significant where b = -1.96, χ2 (1) = 36.60, p < 0.05 implying it had an impact on deaths. Age was also statistically significant where b = 0.196, χ2 (1) = 43.74, p < 0.05. The likelihood of dying is 0.14 times higher in males as compared to females according to the odds ratio Exp (B).

Part II

Discriminant Analysis

Two tests were developed in a firm to determine whether some of the employees will perform in a given position. A sample of 43 employees was examined. The main aim is to group employees as either successful or unsuccessful by using the tests given.

Discriminant analysis suits this case since exclusive grouping was required and the dependent variable was categorical with two groups 0) Unsuccessful and 1) Successful (Bowerman et al., 2019). Two independent variables used in this study (Test 1 and Test 2) were quantitative reporting the scores of the employees in the two tests.

Analysis

Discriminant analysis was carried out in SPSS to classify the employees as successful or unsuccessful based on the two tests. Descriptive statistics were as shown in table 7.

Table 7: Descriptive statistics

Group Statistics
Group	Mean	Std. Deviation	Valid N (listwise)
			Unweighted	Weighted
Unsuccessful	Test1	84.7500	4.24109	20	20.000
	Test2	79.1000	4.38778	20	20.000
Successful	Test1	92.4348	3.47492	23	23.000
	Test2	84.7826	6.23740	23	23.000
Total	Test1	88.8605	5.43175	43	43.000
	Test2	82.1395	6.10847	43	43.000

The mean of test 1 in the unsuccessful group was 84.75 while for test 2 in the unsuccessful group was 79.10. The means for test 1 and 2 in the successful group were 92.43 and 84.78 respectively.

Table 8 shows the importance of the independent variables in the discriminant function used to group the employees.

Table 8: Test of equality of group means

Tests of Equality of Group Means
	Wilks' Lambda	F	df1	df2	Sig.
Test1	.490	42.644	1	41	.000
Test2	.780	11.593	1	41	.001

According to the analysis, both tests scores were statistically significant in for the discriminant function.

Table 9 shows the correlation matrix of the predictor variables.

Table 9: Correlation matrix

Pooled Within-Groups Matrices
	Test1	Test2
Correlation	Test1	1.000	.187
	Test2	.187	1.000

According to the analysis, the correlation between the scores of test 1 and test 2 was r = 0.19. This is a weak positive relationship implying the independent variables are not correlated.

The assumption of multivariate normality was examined and the test results were as shown in the Box’s M statistics given in table 10.

Table 10: Homogeneity of covariance matrix

Test Results
Box's M	5.014
F	Approx.	1.582
	df1	3
	df2	936960.353
	Sig.	.191
Tests null hypothesis of equal population covariance matrices.

According to the analysis, it is clear that groups did not differ in the covariance matrices implying that the assumption is not violated and the analysis can continue.

According to table 11, one discriminant function was found given the two-grouped dependent variable.

Table 11: Canonical discriminant function

Eigenvalues
Function	Eigenvalue	% of Variance	Cumulative %	Canonical Correlation
1	1.161a	100.0	100.0	.733
a. First 1 canonical discriminant functions were used in the analysis.

The strong positive canonical correlation implies that there was a strong association between the discriminant function and the dependent variable (Uurtio et al., 2017).

Table 12 shows the coefficients of the independent variables.

Table 12: Standardized canonical discriminant function coefficients

Standardized Canonical Discriminant Function Coefficients
	Function
	1
Test1	.885
Test2	.328

According to the analysis, Test 1 had the best ability in discriminating as compared to Test 2. This implies that Test 1 is very significant in predicting whether employees will be successful or unsuccessful in the position.

Table 13 shows the unstandardized canonical coefficients of the model.

Table 13: Unstandardized canonical coefficients

Canonical Discriminant Function Coefficients
	Function
	1
Test1	.230
Test2	.060
(Constant)	-25.380
Unstandardized coefficients

The discriminant equation becomes;

D = -25.38 + 0.23*Test 1 + 0.06*Test 2

Table 14 shows the classification of the given variables.

Table 14: Classification

Classification Resultsa,c
		Group	Predicted Group Membership	Total
			Unsuccessful	Successful
Original	Count	Unsuccessful	16	4	20
		Successful	5	18	23
	%	Unsuccessful	80.0	20.0	100.0
		Successful	21.7	78.3	100.0
Cross-validatedb	Count	Unsuccessful	16	4	20
		Successful	5	18	23
	%	Unsuccessful	80.0	20.0	100.0
		Successful	21.7	78.3	100.0
a. 79.1% of original grouped cases correctly classified.
b. 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.
c. 79.1% of cross-validated grouped cases correctly classified.

Analysis showed that 80% of the employees classified as unsuccessful were unsuccessful while 20% who were successful were classified as unsuccessful. 78.30% successful employees were classified as successful while 21.7% successful were classified as unsuccessful. In overall, 79.10% cases were correctly classified.

Reference

Austin, P. C., & Merlo, J. (2017). Intermediate and advanced topics in multilevel logistic regression analysis. Statistics in medicine, 36(20), 3257-3277.

Warner, R. M. (2012). Applied statistics: From bivariate through multivariate techniques. Sage Publications.

Riley, R. D., Van Calster, B., & Collins, G. S. (2021). A note on estimating the Cox‐Snell R 2 from a reported C statistic (AUROC) to inform sample size calculations for developing a prediction model with a binary outcome. Statistics in Medicine, 40(4), 859-864.

Fagerland, M. W., & Hosmer, D. W. (2017). How to test for goodness of fit in ordinal logistic regression models. The Stata Journal, 17(3), 668-686.

Bowerman, B., Drougas, A. M., Duckworth, A. G., Hummel, R. M. Moniger, K. B., & Schur, P. J.  (2019). Business statistics and analytics in practice (9th ed.). McGraw-Hill

Uurtio, V., Monteiro, J. M., Kandola, J., Shawe-Taylor, J., Fernandez-Reyes, D., & Rousu, J. (2017). A tutorial on canonical correlation methods. ACM Computing Surveys (CSUR), 50(6), 1-33.