SPSS Example of a Logistic Regression Analysis - SPSS Help
Consider the following 9-step Hypothesis Testing Procedure:
1. Evaluate the Data
2. Review Assumptions
3. State Hypotheses
4. Select the Test Statistic
5. Distribution of the Test Statistic
6. State the Decision Rule
7. Calculate the Test Statistic
8. Statistical Decision
9. Conclusion
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Exercise 2:
For this exercise please recode the variable corresponding to the depressed state of mind into a dichotomous variable that has two groups: those who rated themselves as rarely depressed are scored 0, and those who rated themselves as sometimes to routinely depressed are scored 1. Using the new variable as the outcome measure, determine which of the following variables increase the odds of being depressed:
1. Smoking history: recoded into currently smoking = 1, and not currently smoking = 0.
2. Gender: Male = 0, Female = 1
3. Quality of life in the past month: recoded so that values 1 to 3 (sometimes to very unhappy) become 0, and 4 to 6 (sometimes to extremely happy) become 1.
4. Total score on the Inventory of Positive Psychological Attitudes scale (IPPA): Enter recoded smoking history and gender in the first block and recoded quality of life and total IPPA score in the second block.
Results for Exercise 2:
A logistic regression was run to answer the research question (n=653). The results are contained in Exercise Figure 13-1. The variables were entered in tow blocks. Smoking status and gender were entered in block 1, which was significant (p=.003), and accounted for 1.8 to 2.4 percent of the variance. The Hosmer and Lemeshow Test indicated a good fit (p=.808). Only smoking made a significant contribution (p=.001). Quality of life and total IPPA score were entered in block 2, which was significant (p=.000). The total model was significant (p=.000), and accounted for 34.3 to 45.7 percent of the variance. The model was a good fit. (Hosmer and Lemeshow, chi square = 4.068, df = 8, p = .851). The sensitivity of the model in predicting depression was 72.3 percent. The specificity in predicting those who were not depressed was 80. Three of the variables, smoking status (p=.032), quality of life (p=.000), and Total IPPA Score (p=.000) were significant predictors. The odds of being depressed were 2 times higher for those who smoked. Higher quality of life was related to lower probability of depression. The IPPA scale is scored from 30 to 210, therefore, a 1 point increase in that score would be of no practical interest. To calculate the effect of a 30 point change in IPPA, multiply the b-weight for TOTAL times 30 and raise 2.1718 to that power (.057 x 30 = 1.71), raising 2.1718 to the power of 1.71 = 5.53. So, for every 30 point increase in the IPPA score, the odds of being depressed go down (negative b-weight) 5.5 times.
Exercise Figure 13-1
LOGISTIC REGRESSION RESULTS
Dependent Variable Encoding
Original Value |
Internal Value |
Rarely |
0 |
Sometimes to routinely |
1 |
BLOCK 1: Method = Enter
Omnibus Tests of Model Coefficients
Chi Square |
df |
Sig | |
Step 1 Step |
11.848 |
2 |
.003 |
Block |
11.848 |
2 |
.003 |
Model |
11.848 |
2 |
.003 |
Model Summary
Step |
-2Log likelihood |
Cox & Snell R Square |
Nagelkerke R Square |
1 |
892.445 |
.018 |
.024 |
Hosmer and Lemeshow Test
Step |
Chi Square |
Df |
Sig. |
1 |
.059 |
1 |
.808 |
Variables in the Equation
B |
S.E. |
Wald |
df |
Sig |
Exp(B) | |
Step 1(a) SMOKEREC |
.903 |
.273 |
10.978 |
1 |
.001 |
2.468 |
Gender |
.169 |
.166 |
1.037 |
1 |
.309 |
1.184 |
Constant |
-.275 |
.137 |
4.039 |
1 |
.044 |
.759 |
a Variables entered on step1: SMOKERERC, GENDER
Block 2: Method = Enter
Omnibus Tests of Model Coefficients
Chi Square |
df |
Sig. | |
Step 1 Step |
262.275 |
2 |
.000 |
Block |
262.275 |
2 |
.000 |
Model |
274.123 |
4 |
.000 |
Model Summary
Step |
-2 Log likelihood |
Cox & Snell R Square |
Nagelkerke R Square |
1 |
630.170 |
.343 |
.457 |
Hosmer and Lemeshow Test
Step |
Chi Square |
df |
Sig. |
1 |
4.068 |
8 |
.851 |
Classification Table (a)
Observed |
Predicted Depression recoded rarely Sometimes to routinely |
Percentage correct |
|||
Step 1 Depression Rarely recoded Sometimes to routinely |
273 87 |
66 227 |
80.5 72.3 |
||
Overall percentage |
76.6 |
||||
a The cut value is .500
Variables in the Equation
B |
S.E. |
Wald |
df |
Sig |
Exp(B) | |
Step 1(a) SMOKEREC |
.771 |
.359 |
4.621 |
1 |
.032 |
2.162 |
GENDER |
.225 |
.206 |
1.187 |
1 |
.276 |
1.252 |
QOLREC |
-1.066 |
.292 |
13.320 |
1 |
.000 |
.344 |
TOTAL |
-.057 |
.005 |
117.961 |
1 |
.000 |
.944 |
Constant |
9.433 |
.832 |
128.681 |
1 |
.000 |
12499.450 |
a Variable(s) entered on step 1: QOLREC, TOTAL
Solution:
Step1. Evaluate the data
The data comes from four independent variables:
- The smoking history, with two levels: currently smoking = 1, and not currently smoking = 0.
- Gender: male = 0, female =1.
- Quality of life in the past month, with two levels: sometimes to very unhappy = 0, sometimes to extremely happy = 1
- Total IPPA Score: From 30 to 210.
The response variable was the categorical variable “depressed”, with twp levels: rarely depressed = 0, sometimes to routinely depressed = 1.
The sample size is \(n=653\).
Step2. Review Assumptions
Given the nature of the problem, we use Logit Regression. The variables were entered in two blocks, smoking story and gender in block 1, and previous quality of life and total IPPA score in block 2. Both block were significant (\(p=0.003\) and \(p=0.000\) respectively)
Step3. State hypothesis
We have to define our null an alternative hypothesis. Our basic questions are:
- What variables are significant predictors? For this question we pose the null hypothesis that the respective variable is not significant.
- Is the overall model significant?
The logit regression will give an answer to all of these questions.
Step4. Select the Statistic.
For testing the significance of the independence variables, we use the Wald’s statistic for. The Wald’s statistic is defined as:
\[Wald=\frac{{{{\hat{\beta }}}^{2}}}{S{{E}^{2}}}\]For the overall goodness of fit we use the Hosmer-Lemeshow test.
Step5. Distribution of the test’s Statistic.
The Wald’s statistic has a Chi-Square distribution, while the Hosmer-Lemeshow test has a Chi-Square distribution.
Step6. State Decision Rule
For the non-significance of the independent variables, we have the following rejection area:
\[W>{{W}_{\alpha }}\]where \({{F}_{\alpha }}\) correspond to the critical level for the Chi-Square distribution, with the respective number of degrees of freedom, for the given level of significance $\alpha $. On the other hand, we reject the Hosmer-Lemeshow test if the p-value is small.
Step7. Calculate Test Statistic
For the smoking variable we have that:
\[W=4.621\] \[p=0.032\]For the gender variable we have that:
\[W=1.187\] \[p=0.276\]For the last month quality of life variable we have that:
\[W=13.320\] \[p=0.000\]For the IPPA variable we have that:
\[W=117.961\] \[p=0.000\]The overall significance (Hosmer-Lemeshow) for the first block is p=0.808, and for the second block is \(p=0.851\).
Step8. Statistical result
The overall fit is reasonably good, and the significant predictors are SMOKING, past month QUALITY of life, and IPPA score.
Step9. Conclusion
Based on the results of our logit regression, we conclude that the overall fit is reasonably good, and the significant predictors are SMOKING, past month QUALITY of life, and IPPA score. Also, we conclude that smokers have 2.162 times higher odds than non-smokers to be depressed. Also, a 30-point change in IPPA leads to a decrease of 5.53 times the odds of being depressed.
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