Cox Proportional Hazard Regression Analysis
There are basically two statistical methods to analyze survival rates (Austin et al., 2016). The Kaplan-Meier method and log-rank test, and Cox proportional hazard regression analysis. The Kaplan-Meier method and log-rank test are non-parametric statistics that justify the survival function (Lacny et al., 2018). Cox proportional hazard regression analysis is a technique for determining the relationship between variables and the survival rate. The measurement of the risk presented for each characteristic is known as the risk ratio. A risk ratio presented by 1 means that all participants have equal risk. Moreover, a risk ratio higher than 1 indicates an increased risk, whereas a risk lesser than one indicates a lesser risk. A ratio can also present Cox proportional analysis. A ratio of 4.5 indicates that a patient with a variable of 4.5 times has a higher chance of being studied. The confidence interval can also be included with risk ratios. The cox proportional hazard analysis is normally presented in a table or graph based on the requirements. The cox model is a statistical technique that provides results based on survival time that has an outcome on one or more predictors. A response variable is used to state the probability of an event such as death occurred before time. Additionally, Cox proportional hazard models state two assumptions: that different strata for survival curves must include hazard functions that are considered proportional over time. The second assumption is that log hazard, and each covariate relationship is linear.
Additionally, Cox proportional survival analysis involves censoring some patients who may not be interested in the study. The event of occurring may not be known of whether the patient survived. Cox proportional are derived from underlying issues based on hazards being tested. In the Framingham heart study dataset provide, we have a survival curve data provided. Cox proportional hazards regression analysis is performed to determine the survival time (risk of dying) where patients are divided into treatment groups. The data is divided into four groups: the subject, serial time in years, status at the serial time given by the event and censored (1=event; 0=censored), and group presented by chemo 1 or placebo 2). The subject’s test includes 12 participants with different survival rates. The model is performed on an excel spread sheet. The null hypothesis (H0) is the risk of dying is not related to patient treatment groups. The alternative hypothesis is the risk of dying is related to the patient treatment groups. The significance level is being conducted at 0.05.
| Variable | Observations | Obs. with missing data | Obs. without missing data | Minimum | Maximum | Mean | Std. deviation |
| Serial Time (years) | 12 | 0 | 12 | 0.500 | 5.000 | 2.396 | 1.557 |
| Status At Serial Time (1=event; 0=censored) | 12 | 0 | 12 | 0.000 | 1.000 | 0.833 | 0.389 |
|
Summary statistics (Qualitative data): | ||||
| Variable | Categories | Counts | Frequencies | % |
| Group (1 Chemo or 2 Placebo) | 1 | 6 | 6 | 50.000 |
| 2 | 6 | 6 | 50.000 |
| Summary statistics (Events): | |||
| Total observed | Total failed | Total censored | Time steps |
| 12 | 10 | 2 | 10 |
| The goodness of fit statistics: | ||
| Statistic | Independent | Full |
| Observations | 10.000 | 10.000 |
| DF | 0.000 | 1.000 |
| -2 Log(Likelihood) | 36.335 | 33.653 |
| AIC | 36.335 | 35.653 |
| SBC | 36.335 | 35.955 |
| Iterations | 1.000 | 3.000 |
| Regression coefficients: | |||||||
| Variable | Value | Standard error | Wald Chi-Square | Pr > Chi² | Hazard ratio | Hazard ratio Lower bound (95%) | Hazard ratio Upper bound (95%) |
| Group (1 Chemo or 2 Placebo)-2 | 1.200 | 0.748 | 2.575 | 0.109 | 3.319 | 0.767 | 14.370 |
| Proportionality test: | |||
| Variable | rho | Chi-square | Pr > Chi² |
| Group (1 Chemo or 2 Placebo)-2 | 0.07205982 | 0.04749483 | 0.827 |
| Global | 0.04749483 | 0.827 |
Conclusion
Survival analysis methods assess different risk factors associated with multiple regression analysis (Katzman et al., 2018). The above graph of the survival distribution function can be used to interpret the results. An observation is made from the graph that the survival rate is 85% at 0.5 years. The survival rate reduces with an increase in years. The survival rate is 5% in the 5th year. Normally, a flat survival curve depicts good survival while a curve that drops towards the point 0 describes poor survival. Survival curves are generally shown as a step function. These models are essential in the statistical analysis of data in public health (Sullivan, 2018). The models interpret survival analysis to determine if certain mechanisms are working. In the survival curve, cox proportional analysis is used to determine whether the subjects survived or died. The model further analyzes the effect of chemo and the placebo of survival subjects. The outcome of the data will assist the researchers in determining statistical significance of the data. The results and the curve will be useful in the analysis to determine whether to increase or decrease the variables. This information can be used in public health to determine whether participants in a study responded well. Using the p-value of 0.04, we fail to reject the alternative hypothesis that the risk of dying is related to the patient treatment group. We conclude that the risk of dying will increase depending on the group the participant is placed.
References
Austin, P. C., Lee, D. S., & Fine, J. P. (2016). Introduction to the analysis of survival data in the
Presence of competing risks. Circulation, 133(6), 601-609.
Katzman, J. L., Shaham, U., Cloninger, A., Bates, J., Jiang, T., & Kluger, Y. (2016). Deep
survival: A deep cox proportional hazards network. stat, 1050(2).
Lacny, S., Wilson, T., Clement, F., Roberts, D. J., Faris, P., Ghali, W. A., & Marshall, D. A.
(2018). Kaplan–Meier survival analysis overestimates cumulative incidence of health-related events in competing risk settings: a meta-analysis. Journal of clinical epidemiology, 93, 25-35.
Sullivan, L. M. (2018). Essentials of biostatistics in public health (3rd ed.). Jones & Bartlett:
Boston, MA. ISBN-13:9781284108194