Drug test
Impact of age and drug on the duration
In this test, a drug was administered to 28 out of the 48 respondents. The age of the respondents and the drug were the independent variables while the participants’ months to death or end of trial or study time was the dependent variable. The observations were recorded for the 48 participants and the data was regressed using the Gretl software. To determine the impact of age and drug on the duration, we analyze the regression output. The measures of interest are the coefficients of the two variables, Z statistics, and the corresponding p-values. The interpretation involves testing the hypothesis that the variables are not statistically significant (the coefficients are zero).
Null hypothesis, H0: X1 = 0, X2 = 0
Alternative hypothesis, HA: X1 ≠ 0, X2 ≠ 0
The coefficient of the drug is 6.2 and 6.58089 for the Weibull and Lognormal models respectively. This indicates that there is a positive relationship between the drug and the duration/study time. This implies that the participants who were administered the drug had longer durations than those who were not administered with the drug. The p-value of the coefficient is less than 0.05 in both models. As shown by the by the stars indicated on the regression output, the coefficient is statistically significant hence we reject the null hypothesis. Therefore we conclude that the drug increases the duration or participant’s months before death or end of the trial.
The coefficient for age is – 0.0 and -0.083399 for the Weibull and Lognormal models respectively. The negative sign indicates a negative relationship between age and duration. Thus, a participant with a higher age is likely to have a shorter duration to death or end of the trial than a participant with a lower age. The p-value of age is less than 0.000 in both models and they are statistically significant as shown by the stars. Thus, we reject the null hypothesis and conclude that the coefficient of age is statistically significant. Therefore, there is a significant inverse relationship between age and duration.
Impact of age and drug on the hazard function (rate of death)
As explained above, the drug has a positive impact on the duration a participant lives before death. Thus, it has an inverse relationship with the hazard function. Therefore, administering the drug to participants reduces the rate of deaths. On the other hand, age has an inverse relationship with the participants’ duration. It, therefore, has a positive association with the hazard function (rate of death).
The preferable model
Both models give the same results on the relationship between drug and age, and duration. However, the values of regression estimates of the two models are slightly different. The lognormal model uses the logarithms of the values of the variables. In this case, the logarithms are normally distributed. The Weibull model is a flexible model that shows that the probability distribution. It focuses on the failure rate or hazard function of lifetime characteristics of the variable (Menard 194-220).
The best model is one that maximizes the optimal values of the coefficients estimated by the regression model. In this case, we compare the Log-likelihood of the two models. Log-likelihood is a measure of the maximization of the estimated coefficients (Menard 194-220). The Log-likelihood for that Weibull model is -42.931 while that of the Log-normal model is -42.800. The Log-likelihood of the Log-normal model is higher than that of the Weibull model. Thus, the Log-normal model maximizes the optimal values of the estimated coefficients more than the Weibull model. Therefore, the Log-normal model is better than the Weibull model as it gives a better fit. I recommend the use of the Log-normal model.
Conclusion for the drug trial
The coefficients of the drug in the model is positive and statistically significant as shown in the above analysis. We can infer that the drug is effective as it increases the participant’s duration to death or end of the trial. Thus, it reduces the hazard function (rate of deaths). On the other hand, age has a significant inverse relationship with duration. Older people have shorter durations to death.
Ordinal Regression Model
Interpretation of cut1, cut2 and cut3
Cut1 is the cut point of the inferred variable that separates that the low warmth (latent variable) from the middle warmth (Menard 194-220). In this case, cut1 is -2.46536 implying that subjects with lower than -2.46536 are classified as having low warmth.
Cut2 separates the middle from the high (Menard 194-220). Cut2 for this model is -0.630904 implying that subjects or people who had between -2.46536 and -0.630904 were categorized as having middle/moderate warmth. Cut3 is 1.26185 meaning that subjects that had between -0.630904 and 1.26185 were classified as having high warmth.
The coefficient of the male is -0.7333 with a Z statistic of -9.3435 and a corresponding p-value of <0.00001. This indicates a negative relationship between male and the dependent variable. The p-value is very low hence the coefficient is statistically significant.
The coefficient of age in the model is -0.0216655 indicating an inverse relationship between age and the dependent variable. The p-value is <0.00001 indicating that the coefficient if age is statistically significant. Thus, there is a significant statistical association between age and the dependent variable. A 1% increase in age causes a decrease in the dependent variable by 0.0216655%.
Works cited
Menard, Scott W. Logistic Regression: From Introductory To Advanced Concepts And
Applications. 1st ed. Los Angeles: SAGE, 2010. Print.
