# Example of Simple Binary Logistic Regression

A medical researcher wants to know how the dosage level of a new medicine affects the presence of bacteria in adults. The researcher conducts an experiment with 30 patients and 6 dosage levels. For two weeks, the researcher gives one dosage level to 5 patients, another dosage level to another 5 patients, and so on. At the end of the two-week period, each patient is tested to determine whether any bacteria are detected.

Because the data include a binary response and one continuous predictor, the researcher uses a binary fitted line plot to determine whether the dosage of a medicine is related to the presence of bacteria.

1. Open the sample data, BacteriaMedicine.MTW.
2. Open the Simple Binary Logistic Regression dialog box.
• Mac: Statistics > Regression > Simple Binary Logistic Regression
• PC: STATISTICS > Binary Logistic > Simple Binary Logistic Regression
3. From the drop-down list, select Response in event/trial format.
4. In Event name, type No bacteria.
5. In Number of events, enter No Bacteria.
6. In Number of trials, enter Trials.
7. In Predictor, enter Dose (mg).
8. On the Graphs tab, select Residual plots.
9. Click OK.

## Interpret the results

The p-value for the medicine dosage is less than the significance level of 0.05. These results indicate that the relationship between the dosage of medicine and the presence of bacteria is statistically significant. The binary fitted line plot shows that as the amount of dosage increases, the likelihood that no bacteria is present increases. Furthermore, the odds ratio indicates that for every 1 mg increase in the dosage level, the likelihood that no bacteria is present increases by approximately 38 times. The fitted line plot and the residual plots show that the model fits the data well.
 Regression Equation
 P(No bacteria) = exp(Y')/(1 + exp(Y')) Y' = −5.247 + 3.626 Dose (mg)
 Response Information
 Variable Value Count Event Name No Bacteria Event 18 No bacteria Non-event 12 Trials Total 30
 Deviance Table
 Source DF Adj Dev Adj Mean Chi-Square P-Value Regression 1 22.7052 22.7052 22.71 <0.0001 Dose (mg) 1 22.7052 22.7052 22.71 <0.0001 Error 4 0.9373 0.2343 Total 5 23.6425
 Model Summary
 Deviance R-sq Deviance R-sq(adj) AIC 96.04% 91.81% 21.68
 Coefficients
 Term Coef SE Coef 95% CI Z-Value P-Value Constant -5.247 1.989 (-9.146, -1.349) -2.64 0.0083 Dose (mg) 3.626 1.295 (1.087, 6.165) 2.80 0.0051
 Odds Ratios for Continuous Predictor
 Odds Ratio 95% CI Dose (mg) 37.5511 (2.96, 475.65)
 Goodness-of-Fit Tests
 Test DF Chi-Square P-Value Deviance 4 0.94 0.9192 Pearson 4 0.70 0.9508 Hosmer-Lemeshow 4 0.70 0.9508
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