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Age

Race

Gender

SES Comorbidity

FIGURE 5.3  Conceptual model for study evaluating outcomes in Medicare beneficiaries admitted for gallstone pancreatitis. The out- comes include receipt of cholecystectomy and gallstone-related readmissions. The factors thought to influence each outcome are shown in the model

size, day of the week, hospital teaching status, and patient characteristics would affect cholecystectomy rates and that receipt of cholecystectomy as well as patient characteristics would influence gallstone-related readmission rates.

The Basics of Advanced Statistical Analysis

Before you jump to complex statistical methods, you need to understand your data by performing simple descriptive statis- tics including the frequencies of categorical variables, means, medians, and distributions of continuous data, and univariate comparisons between groups. You then need to consult with a statistician. You must make sure you are using correct sta- tistical methods, understand the assumptions of the statistical methods you are using, and make sure you do not violate the assumptions.

Multivariate Analysis

Multivariate analysis is a method of obtaining a mathematical relationship between an outcome variable and multiple pre- dictor variables. Various forms of regression are commonly used to control for confounding and establish independent associations among predictor variables and outcomes. Multiple regression fits data into a model that defines the outcome (Y) as a function of multiple predictor variables (x1, x2, …, xi), and the regression equation takes many forms depending on whether the outcome variable is categorical or numerical. Its general form is: Y (Outcome) = b0 + b1x1 + b2x2 +  … + bjxj + e, where Y is the outcome, x1 through xj are the covariates (predictors), b0 is the intercept, b1 through bj are coefficients describing the effect of the specific covariate on the outcome, and e is the error term.

Linear regression is used to study the relationship of a continuous variable to a single predictor variable. In an example given by Afifi et al. in Computer-Aided Multivariate Analysis (see selected references), the researcher is evaluat- ing the effect of height on forced expiratory volume (FEV1). The basic regression equation is: FEV1 = b0 + b1 (height in inches).TherelationshipdiscoveredwasFEV1 = −4.087 + 0.118 (height in inches). So for each inch of increased height, FEV1 increases by a factor of 0.118.

However,we know that other factors such as age also affect FEV1. Multiple linear regression allows these variables to be added to the model providing a less biased estimate. When age is added to the model, the result is FEV1 = −2.761 – 0.027 (age) + 0.114 (height). After controlling for age, the FEV1 increases 0.114 with each increase in height.The independent predictor variables in a multiple linear regression can be con- tinuous or categorical.

Logistic regression is commonly used when an outcome variable is categorical. Again, the predictor variables can be continuous or categorical. Logistic regression models model the log of the odds of the outcome variable. The equation is in the form:Logit [odds] = b0 + b1x1 + b2x2 + … + bjxj.In this case,

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TABLE 5.3  Factors predicting cholecystectomy in Medicare benefi- ciaries admitted for gallstone pancreatitis

odds ratios for each factor can be obtained by exponentiating the beta coefficient: OR = ebx. If the OR is equal to one or the 95% confidence interval includes 1, the associated predictor variable does not have a statistically significant association with the outcome variable. Table 5.3 shows a logistic regres- sion model which models the odds of cholecystectomy in patients admitted with gallstone pancreatitis from the previ- ous example.The outcome is dichotomous – receipt of chole- cystectomy vs.no cholecystectomy.Since we are modeling the odds of cholecystectomy, factors with an OR > 1 predict increased patient odds of receiving cholecystectomy and an OR < 1 predicts decreased odds.Age is a continuous variable. For each 5-year increase in age, patients had 17% decreased odds to undergo cholecystectomy.When compared to patients who didn’t have an ERCP, those who did had 53% higher odds of receiving cholecystectomy. Both are statistically sig- nificant since the 95% confidence intervals do not include the null value of 1. The model also controlled for gender, race, comorbidities, regional differences, and hospital characteris- tics, not all shown.

For regression models to control for confounding and selection bias, the predictors and confounders must be known and included in the model. When constructing a regression model, you can put all the factors in your conceptual model in the statistical model and eliminate factors that are not significant­ in stepwise fashion based on statistical tests

(hypothesis is that the b = 0, or the OR = 1). Conversely, you can start with only your relationship of interest (simple regres- sion) and add factors in stepwise fashion. Your model should be based on your conceptual model. Some factors, while not significant,might be known confounders and should be forced into the model (not removed even if not significant).

Time-to-Event Analysis

Time-to-event analyses are used when the time to a specific event, and not only the occurrence of the event, is important. Survival analysis is the most common example. It is not enough to know if a patient died, but how long they lived before the event occurred,as there is a big difference between dying 1 month and 10 years after cancer surgery. The end point of a time-to-event analysis can be any endpoint such as readmission to the hospital, death, reoperation, etc. The Kaplan–Meier product limit method allows patients to enter the cohort at different points in time and have variable follow-up.This method is used when the exact date of an end point is known and event-free survival is calculated at each time point where an event occurs. Once the event occurs, the time from onset of the study to the event is recorded. A patient is censored if the event of interest does not occur during the follow-up period. In Kaplan–Meier analysis, a “survival” curve (time without an event) can be plotted to illustrate the percentage of event-free patients on the y-axis and follow-up time on the x-axis. Fig. 5.4 shows the Kaplan– Meier “survival” curve for patients readmitted after an initial hospitalization for gallstone pancreatitis. Ninety-six percent of patients in the cholecystectomy group did not require readmission over the first 2 years, while only 56% in the no cholecystectomy group did not require readmission. In other words, the gallstone-related readmission rate was higher (44% vs. 4%) in the no cholecystectomy group. Log-rank tests are used to compare differences in survival between two groups of patients.