Question

Cardiac rehabilitation programs offer “information, support and monitoring for return to activities, symptom management, and risk factor modification. The researchers conducted a study to identify among women factors that are associated with participation in such programs. The data are the ages of 185 women discharged from a hospital in Australia who met eligibility criteria involving discharge for myocardial infarction, artery bypass surgery, angioplasty, or stent. We wish to use these data to obtain information regarding the relationship between age (years) and participation in cardiac rehabilitation program (attend = 1, if participated, and attend = 0, if not). We wish also to know if we may use the results or our analysis to predict the likelihood of participation by a woman if we know her age.

The LOGISTIC Procedure

                               Model Information

                 Data Set                      WORK.CARDIAC

                 Response Variable             attend

                 Number of Response Levels     2

                 Model                         binary logit

                 Optimization Technique        Fisher's scoring

                    Number of Observations Read         184

                    Number of Observations Used         184

                                Response Profile

                       Ordered                      Total

                         Value       attend     Frequency

                             1            1            63

                             2            0           121

                        Probability modeled is attend=1.

                            Model Convergence Status

                 Convergence criterion (GCONV=1E-8) satisfied.

                              Model Fit Statistics

                                                  Intercept

                                   Intercept            and

                     Criterion          Only     Covariates

                     AIC             238.480        233.520

                     SC              241.695        239.950

                     -2 Log L        236.480        229.520

                    Testing Global Null Hypothesis: BETA=0

            Test                 Chi-Square       DF     Pr > ChiSq

            Likelihood Ratio         6.9601        1         0.0083

            Score                    6.9895        1         0.0082

            Wald                     6.7083        1         0.0096

                   Analysis of Maximum Likelihood Estimates

                                     Standard          Wald

      Parameter    DF    Estimate       Error    Chi-Square    Pr > ChiSq

      Intercept     1      1.8744      0.9809        3.6518        0.0560

      age           1     -0.0379      0.0146        6.7083        0.0096

                             The LOGISTIC Procedure

                              Odds Ratio Estimates

                                Point          95% Wald

                   Effect    Estimate      Confidence Limits

                   age          0.963       0.936       0.991

         Association of Predicted Probabilities and Observed Responses

               Percent Concordant     64.1    Somers' D    0.309

               Percent Discordant     33.2    Gamma        0.317

               Percent Tied            2.6    Tau-a        0.140

               Pairs                  7623    c            0.655

1. Write the estimated logistic model
2. Does age affect whether you participated or not? (show all necessary work including hypothesis)
3. Explain the estimate of the regression parameter associated with age.
4. Explain the odds ratio associated with age.

Answer 1

x: Age (years)

y: Participation in cardiac rehabilitation program (attend = 1, if participated, and attend = 0, if not)

Intercept: β₀ = 1.8744

β ₁ = -0.0379

a) Write the estimated logistic model

S = Odds ratio = e ^{β0 + β1x}

ln(S) = ln(e ^{β0 + β1x}) = β₀ +β₁x

So, logistic equation is: ln(S) = 1.8744 - 0.0379x

b) Does age affect whether you participated or not?

Null hypothesis- H₀: β₁ = 0, Age has no relationship with participation in program

Alternate hypothesis- H₀: β₁ ≠ 0, Age has relationship with participation in program

Taking p-value from chi-square = 0.0096

Since p-value (0.0096) is less than 0.05, we reject the null hypothesis and conclude that age has a relationship with participation in program.

c) Explain the estimate of the regression parameter associated with age

Coefficient of age = -0.0379

This signifies the expected change in log odds of participation in program for a one-unit increase in age.

Also, e^-0.0379 = 0.096, which means that we expect 90.4% decrease in the odds of participation in program, for a one-unit increase in age.

d) Explain the odds ratio associated with age

S = Odds ratio = e ^{β0 + β1x}

= e^{1.8744 - 0.0379}