Bivariate analysis

* Bivariate analysis studies the relation between 2 variables while assuming that other factors (other associated variables) would remain stationary or are equally distributed between groups. * A bivariate analysis comparing the effects of 2 antihypertensive drugs will not take into consideration the effects of weight gain, age, sex, salt intake, etc..; assuming that this would not compromise the result of the comparison.

A) Type of variable: 1. Quantitative 2. Qualitative. B) Variable distribution: 1. Normal distribution, equality of variance 2. Other distributions

C) Number of groups: 1. Single: the patient is his own control 2. Two independent groups 3. More than 2 groups. D) Number of measurements: 1. Single measurement 2. Two measurements 3. Repeated measurements.

E) Time to event 1. Kaplan Meir analysis 2. Actuarial analysis… Log rank test F) Role of other variables 1. Regression analysis 2. Logistic & multi nominal regression 3. Cox regression analysis.

* Most of the biological variables follow a Normal (or near Normal) distribution. * The means of values usually acquire a Normal distribution when the number of patients is >30 per group. * Qualitative variables will follow a Normal distribution if the expected counts are >5. * A change of scale of (x), into log (x), (1/x) or else may achieve a Normal distribution of the new scale.

* Fortunately enough, the variances of a Normally distributed variable are usually “equal” among compared groups. * Care should be taken not to "twist" variables to reach "Normality" In order to use a "parametric test", as we can always use the –as powerful- distribution-free (non-parametric) tests.

* Normality can be simply checked out by a histogram, verifying number of values in relation to SD. * Statistical software packages offers numerous tests to verify normal distribution e.g. Kolmogorov-Smirnov and Shapiro-Wilk tests for Normality and Levene’s test for equality of variance. A statistically significant result indicates poor fit.

Independent groups of patients 2 qualitative variables Chi- square test 1 Fisher's exact test 2 1 qualitative and 1 quantitative variable Parametric tests (Normal distribution) 3 One measurement Student's test (2 groups) One-way ANOVA (>2 groups) Repeated measurement ANOVA 4. Non-parametric tests (other distributions) Mann & Whitney test (2 groups) Kruskall & Wallis test (>2 groups) 2 quantitative variables Correlation Correlation coefficient 5 Spearman's rank (other distribtuion) Regression 5

One group Analysis Qualitative variable. McNemar's test (2-class variable) 1 Cochran's Q test (k-class variable)2 Quantitative variable Distribution-free tests (other distributions) Paired Wilcoxon or Sign test (one measurement) Friedman's test: 2-Way ANOVA by ranks (repeated measurments) Paired Student's test (Normal distribution 3 )

* The general multivariate equation * G = b 0 + b 1 x 1 + b 2 x 2 + b 3 x 3 …..+ ∑ * Suppose that we are running a hospital and we want to calculate the cost (G) in relation to the number of treated patients (x 1 ), number of surgeries performed (x 2 ), number of laboratory tests (x 3 ), etc… * First there is a basic (b 0 ) running cost (salaries, electricity, rent of equipment, etc…) that should be paid, even if we have no patients at all. * To the latter, we have to add an average cost per patient (b 1 ; regression coefficient), per number of surgeries performed (b 2 ), per laboratory test (b 3 ).

* A multivariate analysis model studies the overall as well as the individual effects of multiple variables on outcome variable. The latter is called the “dependent” variable for being dependent on the former; which by deduction are called “independent” variables or independent “predictors”. * 1) The overall effect is called the “model fit”; which describes how the independent predictors fit-in the model by explaining the variability of outcome. A part of variability will be “left over” unexplained; which is called the residual variance. * A large residual variance, indicates a poor model-fit and calls for the need to introduce other variables into the model to fit-in and explain more variation.

* 2) The individual contribution of independent predictors is based upon the estimation of the regression coefficients (b); each representing the independent contribution of independent variables (x) on outcome (G). The sign and magnitude of the regression coefficient represent the direction and weight of the variable effect on outcome. * G= b 0 + b 1 x 1 + b 2 x 2 + b 3 x 3 …..+ ∑ * As shown in the equation, the outcome (G) is a function of the sum of the multiplication of each independent variable (x) by its corresponding regression coefficient (b); to which is added a basic (flat) rate (b 0 ) where none of the independent variables affect outcome and a constant ( ∑).

OutcomeExampleModelAssumption 1- Binary.Need of revascularization after PCI. logistic regression 1- Each unit increase in a predictor multiplies the odds of the outcome by a certain factor that equals to the predictor OR. 2- The effect of several variables is the multiplicative product of their individual effects. 3- Normal distribution of predictors across outcome categories* 2- Categorical. Need for emergency, urgent and elective need of revascularization after PCI -Discriminate functional analysis 3-Time to event. Time to revascularization after PCI Cox (proportional hazards) regression The ratio of the hazard functions for persons with and without a given risk factor is the same over the entire study period. 3- Continuous Duration of surgery. Multiple (linear) regression analysis. A unit change of independent variables, changes the mean value of outcome