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estimated by this method describe the expected change in response per unit increase in each continuous predictor, independently. When the predictor variable is categorical, the associated parameters represent the expected change in response for each particular category relative to a chosen reference category. For more information and in-depth discussion, the interested reader is referred to Ramsey and Schafer (2002); Ott (1993) and Montgomery et al. (2001).

3. Analysis of categorical data

The chi-squared (x2) test is typically applied when an investigator needs to tell if the categories of one variable occur independently of the categories of another variable. The x2 test also may be utilized for the purpose of testing the goodness of fit of a more complicated statistical model.

Logistic regression is a parametric method used for examining the relationship between a binary response variable (one that is categorical having only two categories) and a set of independent predictor variables that can be either continuous or categorical. Unlike OLS, the natural exponent of an estimated parameter represents the contribution to the odds of response occurring, where we can think of the response as being the more interesting event of the two possible outcomes. For continuous predictor variables, the natural exponent of the estimated parameter represents the contribution to the odds of response per unit increase in the predictor variable. In cases where a predictor is a categorical variable, the natural exponent of the parameter represents the contribution to the odds of the response occurring for that category relative to a reference category chosen by the investigator. For more information and in-depth discussion of these categorical statistical methods see Agresti (1996).

4. Analysis of time-to-event data

In aging studies, researchers may be interested in not only the occurrence of a certain event but also in the timing of the event. This type of data is called time-to-event data. Statistical methods applicable to this type of data are generally known as survival analysis. Time-to-event data have two distinguishing features: censoring and time-dependent covariates (Allison, 1995). For example, suppose that a young investigator gets a one-year pilot grant to follow the effect of a new anti-hypertension agent on the occurrence of heart attack in a certain rodent model. By the end of the one-year period, both the treatment group and the control group have some subjects that had the event (the heart attack) and some had not. The status of animals without events is censored. In this context, censored means that we cannot know whether, much less when, they would have a heart attack after the follow-up period. As another example of censoring data, we may wish to study lifespan and mortality rate by conducting an aging study. For some animal models with a relatively long lifespan, such as nonhuman primates, it is difficult or impractical to follow the animals until death. By the end of the study, one can expect that there will be some subjects with unknown life expectancy (i.e., censored). Moreover, during the follow-up period, the values of some covariates, such as blood pressure and heart function, may also change over time. Conventional statistical methods such as logistic regression and linear regression cannot model the event and the timing of event simultaneously, nor can they incorporate changes in covariates over time. Modern survival analysis methods are capable of handling these complexities inherent in longitudinal time-to-event data analysis (Singer and Willett, 2003).

Survival analysis techniques model the event and timing of the event simultaneously by estimating the hazard function or the survivor function. The hazard function, h(t) is the rate, not the probability, of an event (failure) in a very short time interval. The survivor function, S(t), is the probability that the time to event is greater than t. The hazard function, survivor function, and probability density function of the event time, T, are equivalent. That is, we can derive all of these three functions if we know any one of them (Allison, 1995). Some widely used nonparametric, semiparametric, and parametric survival analysis methods, their assumptions, and the survivor and hazard functions to be estimated are listed in Table 14.3. In summary, the proportional hazards model (Cox, 1972) is the most popular model because, as a semiparametric method, it is the most robust. The nonparametric methods such as KaplanMeier and life-table methods are useful for comparing survival curves and evaluating model fit for regression methods (Allison, 1995).

Given the unique features of survival analysis, there are some specific considerations for sample size calculation for studies using time-to-event data. For example, in addition to the information needed for sample size calculation using conventional methods, the researcher may also need to know the follow-up time, an estimate of the overall follow-up event rate, the underlying assumption of event time, and/or the ratio of median/mean survival time of different groups depending on the study design. The most popular approach for sample size calculation of survival analysis is based on the assumption that the survival times are random draws from an exponential distribution (Schoenfeld and Richer, 1982). Several software packages have implemented either this approach or similar approaches (Iwane et al., 1997). One property of the exponential distribution is that the hazard rate is constant over time, but this is not always true in reality. Some other distributions, such as the Weibull distribution, may be more appropriate to characterize the distribution of survival time especially when the follow-up period is long and the hazard rate is accelerated with time. Heo et al. (1997) developed an approximate sample calculation method for the Weibull regression

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