Medical Statistics: Hypothesis Testing

Medical Statistics: Hypothesis Testing Nimrod Lavi, MD Adhir Shroff, MD, MPH

Agenda • Types of variables • Descriptive statistics • What is a hypothesis • Definition of a p-value • Sample vs. universe • Comparative statistics • T-tests • Chi-square

Continuous variable • One in which research participants differ in degree or amount. • “susceptible to infinite gradations” (p. 176, Pedhazur & Schmelkin, 1991) • Examples: height, weight, age

Categorical variable • Participants belong to, or are assigned to, mutual exclusive groups • Nominal • Used to group subjects • Numbers are arbitrary • Examples: sex, race, dead/alive, marital status • Ordinal (rank) • Given a numerical value in accordance to their rank on the variable • Numerical values assigned to participants tells nothing of the distance between them • Examples: class rank, finishers in a race

Independent “predictor variable” Usually on the “x” axis Dependent “outcome” variable Usually on the “y” axis The independent variable (a treatment) leads to the dependent variable (outcome) Ultimately, we are interested in differences between dependent variables Independent vs Dependent Variable Dependent Independent

Descriptive Statistics • These are measures or variables that summarize a data set • 2 main questions • Index of central tendency (ie. mean) • Index of dispersion (ie. std deviation)

Data set for ICD complications in 2005 14 patients Sex: F, F, M, M, F, F, F, M, F, M, M, F, F, F Make: G, S, G, G, G, M, S,S, G,G, M, S Central tendency is summarized by proportion or frequency Sex: M 5/14 = .36 or 36% F 9/14 = .64 or 64% Make: G 6/12 = .5 or 50% S 4/12 = .33 or 33% M 2/12 = .17 or 17% Dispersion not really used in categorical data Descriptive Statistics • Categorical data

Data set SBP among a group of CHF pts in VA clinic 13 patients 100, 95, 98, 172, 74, 103, 97, 106, 100, 110, 118, 91, 108 Central Tendency Mean mathematical average of all the values Σ (xi+xii…xn)/n Median value that occupies middle rank, when values are ordered from least to greatest Mode Most commonly observed value(s) Descriptive Statistics • Continuous variable

Data set SBP among a group of CHF pts in VA clinic 13 patients 100, 95, 98, 172, 74, 103, 97, 106, 100, 110, 118, 91, 108 Central Tendency Mean mathematical average of all the values Σ (xi+xii…xn)/n = (100+95+98+172+74+103+ 97+106+100+110+118+ 91+108)/13 = 105.5 Descriptive Statistics • Continuous variable

Data set SBP among a group of CHF pts in VA clinic 13 patients 100, 95, 98, 172, 74, 103, 97, 106, 100, 110, 118, 91, 108 Central Tendency Median value that occupies middle rank, when values are ordered from least to greatest 74, 91, 95, 97, 98, 100, 100, 103, 106, 108, 110, 118, 172 Useful if data is skewed or there are outliers Descriptive Statistics • Continuous variable

Data set SBP among a group of CHF pts in VA clinic 100, 95, 98, 172, 74, 103, 97, 106, 100, 110, 118, 91, 108 Index of dispersion Standard deviation measure of spread around the mean Calculated by measuring the distance of each value from the mean, squaring these results (to account for negative values), add them up and take the sq root Descriptive Statistics • Continuous variable

Descriptive Statistics: “Normal”

Descriptive Statistics: Confidence Intervals • “Range of values which we can be confident includes the true value” • Defines the “inner zone” about the central index (mean, proportion or ration) • Describes variability in the sample from the mean or center • Will find CI used in describing the difference between means or proportions when doing comparisons between groups Altman DG. Practical Statistics for Medical Research ;1999

Descriptive Statistics: Confidence Intervals • For example, a “95% CI” indicates that we are 95% confident that the population mean will fall within the range described • Can be used similar to a p-value to determine significant differences • CI is similar to a measure of spread, like SD • As sample size increase or variability in the measurement decrease, the CI will become more narrow

Descriptive Statistics: Confidence Intervals • Prospective, randomized, multicenter trial of different management strategies for ACS • 2500 pts enrolled in Europe with 6 month follow-up • Primary endpoints: Composite endpoint of death and myocardial infarction after 6 months L a n c e t 1999; 3 5 4 : 7 0 8 – 1 5

Descriptive Statistics: Confidence Intervals L a n c e t 1999; 3 5 4 : 7 0 8 – 1 5

Descriptive Statistics: Confidence Intervals L a n c e t 1999; 3 5 4 : 7 0 8 – 1 5 *Risk ratio= Riskinvasive / Risknoninvasive When CI cross 1 or whatever designates equivalency, the p-value not be significant.

Review Calculate: RRR, ARR, NNT RRR = (12.1-9.4) / 12.1 = 22% Descriptive Statistics: Confidence Intervals L a n c e t 1999; 3 5 4 : 7 0 8 – 1 5 ARR = 12.1 - 9.4 = 2.7% NNT = 100 / ARR = 100 / 2.7 = 37

Hypothesis • Statement about a population, where a certain parameter takes a particular numerical value or falls in a certain range of values. • Examples: • A director of an HMO hypothesizes that LOS p AMI is longer than for CHF exacerbation • An investigator states that a new therapy is 10% better than the current therapy • Bivalirudin is not-inferior to heparin/eptifibitide for coronary PCI

Null Hypothesis (Ho) • “Innocent until proven guilty” • Null hypothesis (Ho) usually states that no difference between test groups really exists • Fundamental concept in research is the concept of either “rejecting” or “conceding” the Ho • State the Ho: • A director of an HMO hypothesizes that LOS p AMI is longer than for CHF exacerbation • An investigator states that a new therapy is 10% better than the current therapy • Bivalirudin is not-inferior to heparin/eptifibitide for PCI

Null Hypothesis (Ho):Courtroom Analogy • The null hypothesis is that the defendant is innocent. • The alternative is that the defendant is guilty. • If the jury acquits the defendant, this does not mean that it accepts the defendant’s claim of innocence. • It merely means that innocence is plausible because guilt has not been established beyond a reasonable doubt. Graduate Workshop in Statistics Session 4. Hamidieh K. 2006 Univ of Michigan

Extrapolation of Research Findings • Sample population vs. the world • If your study shows that treatment A is better than treatment B • You cannot conclude that treatment A is ALWAYS better than treatment B • You only sampled a small portion of the entire population, so there is always a chance that your observation was a chance event

Extrapolation of Research Findings • At what point are we comfortable concluding that there is a difference between the groups in our sample • In other words, what is the false-positive rate that we are willing to accept • What is this called in statistical terms?

Definition of p-value • With any research study, there is a possibility that the observed differences were a chance event • The only way to know that a difference is really present with certainty, the entire population would need to be studied • The research community and statisticians had to pick a level of uncertainty at which they could live

Definition of p-value • This level of uncertainty is called type 1 error or a false-positive rate

“Power” Stay tuned…. Two Types of Errors Trt has no effect Trt has an effect Graduate Workshop in Statistics Session 4. Hamidieh K. 2006 Univ of Michigan

Definition of p-value • This level of uncertainty is called type 1 error or a false-positive rate (a) • More commonly called a p-value • Statistical significance will be recognized if p≤ 0.05 (can be set lower if one wishes)

Trade-Off in Probability for Two Errors There is an inverse relationship between the probabilities of the two types of errors. Increase probability of a type I error → decrease in probability of a type II error .01 .05 Graduate Workshop in Statistics Session 4. Hamidieh K. 2006 Univ of Michigan

Definition of p-value • This level of uncertainty is called type 1 error or a false-positive rate (a) • More commonly called a p-value • In general, p≤ 0.05 is the agreed upon level • In other words, the probability that the difference that we observed in our sample occurred by chance is less than 5% • Therefore we can reject the Ho

Definition of p-value Stating the Conclusions of our Results • When the p-value is small, we reject the null hypothesis or, equivalently, we accept the alternative hypothesis. • “Small” is defined as a p-value  a, where a = acceptable false (+) rate (usually 0.05). • When the p-value is not small, we conclude that we cannot reject the null hypothesis or, equivalently, there is not enough evidence to reject the null hypothesis. • “Not small” is defined as a p-value > a, where a = acceptable false (+) rate (usually 0.05). Graduate Workshop in Statistics Session 4. Hamidieh K. 2006 Univ of Michigan

Agenda • Types of variables • Descriptive statistics • What is a hypothesis • Definition of a p-value • Sample vs. universe • Comparative statistics • t-tests • Chi-square

Two Sample Tests: Continuous Variable • t-test • Comparing two groups, statistical significance is determined by: • Magnitude of the observed difference • Bigger differences are more likely to be significant • Spread, or variability, of the data • Larger spread will make the differences not significant

Two Sample Tests: Continuous Variable

Two Sample Tests: Continuous Variable • t-test • Comparing two groups, statistical significance is determined by: • Magnitude of the observed difference • Bigger differences are more likely to be significant • Spread, or variability, of the data • Larger spread will make the differences not be significant • Key is to compare the difference between groups with the variability within each group

Two Sample Tests: Continuous Variable • Types t-tests • Student t-test or two sample t-test • Used if independent variables are unpaired • Example: • A randomized trial to high dose statin versus placebo post AMI • Paired t-test • Used if independent variables are paired • Each person is measured twice under different conditions • Similar individuals are paired prior to an experiment • Each receives a different trt, same response is measured • Example: • A study of ejection fraction in patients before and after Bi-V pacing

Two Sample Tests: Continuous Variable • t-test • Tails • “Two-tailed” • Most commonly used in clinical research studies • Means that the treatment group can be better or worse than the control group • “One-tailed” • Used only if the groups can only differ in one direction

What type of test should be run? How are the data related or are they? Data entered into a statistical program… p value = 0.2329, not significant Example: t-test

Two Sample Tests: Categorical Variables • Chi square (χ2) analysis • Data that is organized into frequency, generate proportions • Based on comparing what values are expected from the null hypothesis to what is actually observed • Greater the difference between the observed and expected, the more likely the result will be significant

- + Chi square (χ2) analysis Outcome Therapy Totals a+b+c+d • Null hypothesis states that outcomes of therapy A and B are equally successful • This is how the expected outcomes are determined

- + Chi square (χ2) analysis Outcome Therapy Totals a+b+c+d • Next the actual observed values are then recorded • With this information the χ2 value can be calculated and a p-value will be generated

Arrange data into a 2x2 table Treatment groups along the vertical axis, Outcomes alone the horizontal axis Example: χ2 analysis

Data entered into a statistical program P-value 0.6392 Not a significant difference Example: χ2 analysis

Example: Ear Infections and Xylitol Experiment: n = 533 children randomized to 3 groups Group 1: Placebo Gum; Group 2: Xylitol Gum; Group 3: Xylitol Lozenge Response = Did child have an ear infection? Group Infection Count 1 placebo Y 49 2 gum N 150 3 lozenge Y 39 4 placebo N 129 5 gum Y 29 6 lozenge N 137 Graduate Workshop in Statistics Session 5. Hamidieh K. 2006 Univ of Michigan

Medical Statistics: Hypothesis Testing