Statistical Methods in Clinical Research James B. Spies M.D., MPH Professor of Radiology Georgetown University School of Medicine Washington, DC

Overview Data types Summarizing data using descriptive statistics Standard error Confidence Intervals

Overview P values One vs two tailed tests Alpha and Beta errors Sample size considerations and power analysis Statistics for comparing 2 or more groups with continuous data Non-parametric tests

Overview Regression and Correlation Risk Ratios and Odds Ratios Survival Analysis Cox Regression

Further Study Medical Statistics Made Easy M. Harris and G. Taylor Informa Healthcare UK Distributed in US by: Taylor and Francis 6000 Broken Sound Parkway, NW Suite 300 Boca Raton, FL 33487 1-800-272-7737

Types of Data Discrete Data-limited number of choices Binary: two choices (yes/no) Dead or alive Disease-free or not Categorical: more than two choices, not ordered Race Age group Ordinal: more than two choices, ordered Stages of a cancer Likert scale for response E.G. strongly agree, agree, neither agree or disagree, etc.

Types of data Continuous data Theoretically infinite possible values (within physiologic limits) , including fractional values Height, age, weight Can be interval Interval between measures has meaning. Ratio of two interval data points has no meaning Temperature in celsius, day of the year). Can be ratio Ratio of the measures has meaning Weight, height

Types of Data Why important? The type of data defines: The summary measures used Mean, Standard deviation for continuous data Proportions for discrete data Statistics used for analysis: Examples: T-test for normally distributed continuous Wilcoxon Rank Sum for non-normally distributed continuous

Descriptive Statistics Characterize data set Graphical presentation Histograms Frequency distribution Box and whiskers plot Numeric description Mean, median, SD, interquartile range

Histogram Continuous Data No segmentation of data into groups

Frequency Distribution Segmentation of data into groups Discrete or continuous data

Box and Whiskers Plots

Box and Whisker Plots Popular in Epidemiologic Studies Useful for presenting comparative data graphically

Numeric Descriptive Statistics Measures of central tendency of data Mean Median Mode Measures of variability of data Standard Deviation Interquartile range

Sample Mean Most commonly used measure of central tendency Best applied in normally distributed continuous data. Not applicable in categorical data Definition: Sum of all the values in a sample, divided by the number of values.

Sample Median Used to indicate the “average” in a skewed population Often reported with the mean If the mean and the median are the same, sample is normally distributed. It is the middle value from an ordered listing of the values If an odd number of values, it is the middle value If even number of values, it is the average of the two middle values. Mid-value in interquartile range

Sample Mode Infrequently reported as a value in studies. Is the most common value More frequently used to describe the distribution of data Uni-modal, bi-modal, etc.

Interquartile range Is the range of data from the 25th percentile to the 75th percentile Common component of a box and whiskers plot It is the box, and the line across the box is the median or middle value Rarely, mean will also be displayed.

Standard Error A fundamental goal of statistical analysis is to estimate a parameter of a population based on a sample The values of a specific variable from a sample are an estimate of the entire population of individuals who might have been eligible for the study. A measure of the precision of a sample in estimating the population parameter.

Standard Error Standard error of the mean Standard deviation / square root of (sample size) (if sample greater than 60) Standard error of the proportion Square root of (proportion X 1 - proportion) / n) Important: dependent on sample size Larger the sample, the smaller the standard error.

Clarification Standard Deviation measures the variability or spread of the data in an individual sample. Standard error measures the precision of the estimate of a population parameter provided by the sample mean or proportion.

Standard Error Significance: Is the basis of confidence intervals A 95% confidence interval is defined by Sample mean (or proportion) ± 1.96 X standard error Since standard error is inversely related to the sample size: The larger the study (sample size), the smaller the confidence intervals and the greater the precision of the estimate.

Confidence Intervals May be used to assess a single point estimate such as mean or proportion. Most commonly used in assessing the estimate of the difference between two groups.

Confidence Intervals Commonly reported in studies to provide an estimate of the precision of the mean.

Confidence Intervals

P Values The probability that any observation is due to chance alone assuming that the null hypothesis is true Typically, an estimate that has a p value of 0.05 or less is considered to be “statistically significant” or unlikely to occur due to chance alone. The P value used is an arbitrary value P value of 0.05 equals 1 in 20 chance P value of 0.01 equals 1 in 100 chance P value of 0.001 equals 1 in 1000 chance.

P Values and Confidence Intervals P values provide less information than confidence intervals. A P value provides only a probability that estimate is due to chance A P value could be statistically significant but of limited clinical significance. A very large study might find that a difference of .1 on a VAS Scale of 0 to 10 is statistically significant but it may be of no clinical significance A large study might find many “significant” findings during multivariable analyses. “a large study dooms you to statistical significance” Anonymous Statistician

P Values and Confidence Intervals Confidence intervals provide a range of plausible values of the population mean For most tests, if the confidence interval includes 0, then it is not significant. Ratios: if CI includes 1, then is not significant The interval contains the true population value 95% of the time. If a confidence interval range is very wide, then plausible value might range from very low to very high. Example: A relative risk of 4 might have a confidence interval of 1.05 to 9, suggesting that although the estimate is for a 400% increased risk, an increased risk of 5% to 900% is plausible.

Errors Type I error Claiming a difference between two samples when in fact there is none. Remember there is variability among samples- they might seem to come from different populations but they may not. Also called the  error. Typically 0.05 is used

Errors Type II error Claiming there is no difference between two samples when in fact there is. Also called a  error. The probability of not making a Type II error is 1 - , which is called the power of the test. Hidden error because can’t be detected without a proper power analysis

Alternative Hypothesis Errors Test Result Null Hypothesis H0 Alternative Hypothesis H1 No Error Type I  Type II  Truth

Sample Size Calculation Also called “power analysis”. When designing a study, one needs to determine how large a study is needed. Power is the ability of a study to avoid a Type II error. Sample size calculation yields the number of study subjects needed, given a certain desired power to detect a difference and a certain level of P value that will be considered significant. Many studies are completed without proper estimate of appropriate study size. This may lead to a “negative” study outcome in error.

Sample Size Calculation Depends on: Level of Type I error: 0.05 typical Level of Type II error: 0.20 typical One sided vs two sided: nearly always two Inherent variability of population Usually estimated from preliminary data The difference that would be meaningful between the two assessment arms.

One-sided vs. Two-sided Most tests should be framed as a two-sided test. When comparing two samples, we usually cannot be sure which is going to be be better. You never know which directions study results will go. For routine medical research, use only two-sided tests.

Sample size for proportions Stata input: Mean 1 = .2, mean 2 = .3,  = .05, power (1-) =.8.

Sample Size for Continuous Data Stata input: Mean 1 = 20, mean 2 = 30,  = .05, power (1-) =.8, std. dev. 10.

Statistical Tests Parametric tests Non-parametric tests Continuous data normally distributed Non-parametric tests Continuous data not normally distributed Categorical or Ordinal data

Comparison of 2 Sample Means Student’s T test Assumes normally distributed continuous data. T value = difference between means standard error of difference T value then looked up in Table to determine significance

Paired T Tests Uses the change before and after intervention in a single individual Reduces the degree of variability between the groups Given the same number of patients, has greater power to detect a difference between groups

Analysis of Variance Used to determine if two or more samples are from the same population- the null hypothesis. If two samples, is the same as the T test. Usually used for 3 or more samples. If it appears they are not from same population, can’t tell which sample is different. Would need to do pair-wise tests.

Non-parametric Tests Testing proportions Testing ordinal variables (Pearson’s) Chi-Squared (2) Test Fisher’s Exact Test Testing ordinal variables Mann Whiney “U” Test Kruskal-Wallis One-way ANOVA Testing Ordinal Paired Variables Sign Test Wilcoxon Rank Sum Test

Use of non-parametric tests Use for categorical, ordinal or non-normally distributed continuous data May check both parametric and non-parametric tests to check for congruity Most non-parametric tests are based on ranks or other non- value related methods Interpretation: Is the P value significant?

(Pearson’s) Chi-Squared (2) Test Used to compare observed proportions of an event compared to expected. Used with nominal data (better/ worse; dead/alive) If there is a substantial difference between observed and expected, then it is likely that the null hypothesis is rejected. Often presented graphically as a 2 X 2 Table

Chi-Squared (2) Test Chi-Squared (2) Formula Not applicable in small samples If fewer than 5 observations per cell, use Fisher’s exact test

Correlation Assesses the linear relationship between two variables Example: height and weight Strength of the association is described by a correlation coefficient- r r = 0 - .2 low, probably meaningless r = .2 - .4 low, possible importance r = .4 - .6 moderate correlation r = .6 - .8 high correlation r = .8 - 1 very high correlation Can be positive or negative Pearson’s, Spearman correlation coefficient Tells nothing about causation

Correlation Source: Harris and Taylor. Medical Statistics Made Easy

Correlation Perfect Correlation Source: Altman. Practical Statistics for Medical Research

Correlation Correlation Coefficient .3 Correlation Coefficient 0 Source: Altman. Practical Statistics for Medical Research

Correlation Correlation Coefficient -.5 Correlation Coefficient .7 Source: Altman. Practical Statistics for Medical Research

Regression Based on fitting a line to data Provides a regression coefficient, which is the slope of the line Y = ax + b Use to predict a dependent variable’s value based on the value of an independent variable. Very helpful- In analysis of height and weight, for a known height, one can predict weight. Much more useful than correlation Allows prediction of values of Y rather than just whether there is a relationship between two variable.

Regression Types of regression Linear- uses continuous data to predict continuous data outcome Logistic- uses continuous data to predict probability of a dichotomous outcome Poisson regression- time between rare events. Cox proportional hazards regression- survival analysis.

Multiple Regression Models Determining the association between two variables while controlling for the values of others. Example: Uterine Fibroids Both age and race impact the incidence of fibroids. Multiple regression allows one to test the impact of age on the incidence while controlling for race (and all other factors)

Multiple Regression Models In published papers, the multivariable models are more powerful than univariable models and take precedence. Therefore we discount the univariable model as it does not control for confounding variables. Eg: Coronary disease is potentially affected by age, HTN, smoking status, gender and many other factors. If assessing whether height is a factor: If it is significant on univariable analysis, but not on multivariable analysis, these other factors confounded the analysis.

Risk Ratios Risk is the probability that an event will happen. Number of events divided by the number of people at risk. Risks are compared by creating a ratio Example: risk of colon cancer in those exposed to a factor vs those unexposed Risk of colon cancer in exposed divided by the risk in those unexposed.

Risk Ratios Typically used in cohort studies Prospective observational studies comparing groups with various exposures. Allows exploration of the probability that certain factors are associated with outcomes of interest For example: association of smoking with lung cancer Usually require large and long-term studies to determine risks and risk ratios.

Interpreting Risk Ratios A risk ratio of 1 equals no increased risk A risk ratio of greater than 1 indicates increased risk A risk ratio of less than 1 indicates decreased risk 95% confidence intervals are usually presented Must not include 1 for the estimate to be statistically significant. Example: Risk ratio of 3.1 (95% CI 0.97- 9.41) includes 1, thus would not be statistically significant.

Odds Ratios Odds of an event occurring divided by the odds of the event not occurring. Odds are calculated by the number of times an event happens by the number of times it does not happen. Odds of heads vs the odds of tails is 1:1 or 1.

Odds Ratios Are calculated from case control studies Case control: patients with a condition (often rare) are compared to a group of selected controls for exposure to one or more potential etiologic factors. Cannot calculate risk from these studies as that requires the observation of the natural occurrence of an event over time in exposed and unexposed patients (prospective cohort study). Instead we can calculate the odds for each group.

Comparing Risk and Odds Ratios For rare events, ratios very similar If 5 of 100 people have a complication: The odds are 5/95 or .0526. The risk is 5/100 or .05. If more common events, ratios begin to differ If 30 of 100 people have a complication: The odds are 30/70 or .43 The risk is 30/100 or .30 Very common events, ratios very different Male versus female births The odds are .5/.5 or 1 The risk is .5/1 or .5

Risk reduction Absolute risk reduction: amount that the risk is reduced. Relative risk reduction: proportion or percentage reduction. Example: Death rate without treatment: 10 per 1000 Death rate with treatment: 5 per 1000 ARR = 5 per 1000 RRR = 50%

Survivial Analysis Evaluation of time to an event (death, recurrence, recover). Provides means of handling censored data Patients who do not reach the event by the end of the study or who are lost to follow-up Most common type is Kaplan-Meier analysis Curves presented as stepwise change from baseline There are no fixed intervals of follow-up- survival proportion recalculated after each event.

Survival Analysis Source: Altman. Practical Statistics for Medical Research

Kaplan-Meier Curve Source: Wikipedia

Kaplan-Meier Analysis Provides a graphical means of comparing the outcomes of two groups that vary by intervention or other factor. Survival rates can be measured directly from curve. Difference between curves can be tested for statistical significance.

Cox Regression Model AKA: Proportional Hazards Survival Model. Used to investigate relationship between an event (death, recurrence) occurring over time and possible explanatory factors. Reported result: Hazard ratio (HR). Ratio of the hazard in one group divided the hazard in another. Interpreted same as risk ratios and odds ratios HR 1 = no effect HR > 1 increased risk HR < 1 decreased risk

Cox Regression Model Common use in long-term studies where various factors might predispose to an event. Example: after uterine embolization, which factors (age, race, uterine size, etc) might make recurrence more likely.

Summary Understanding basic statistical concepts is central to understanding the medical literature. Not important to understand the basis of the tests or the underlying math. Need to know when a test should be used and how to interpret its results