nd on the basis of our study we conclude that there is a difference (cell d), this too is a correct decision. In the best of all worlds, all of the possibilities would fall into one of these two cells. Unfortunately, this is rarely, if ever, the case. There are times when there is no difference between the therapies, but on the basis of the samples of subjects included in our study, we erroneously conclude that they differ (cell c). This is called a type I error. It is also possible that there really is a difference between the therapies, but on the basis of the samples included in our study we erroneously conclude that there is no difference (cell b); this is called a type II error. (In this situation, the therapies differ, but we fail to detect the difference in our study samples.) The probability that we will make a type I error is designated α, and the probability that we will make a type II error is designated β.
α is the so-called P value, which is seen in many published papers and has been sanctified by many years of use. When you see " P < .05," the reference is to α. What does P < .05 mean? It tells us that we have concluded that therapy A differs from therapy B on the basis of the sample of subjects included in our study, which we found to differ. The probability that such a difference could have arisen by chance alone, and that this difference between our groups does not reflect any true difference between therapies A and B, is only .05 (or 1 in 20).
How do these concepts help us to arrive at an estimate of the sample size that we need? If we ask the question, "How many people do we have to study in a clinical trial?" we must be able to specify a number of items
Use the same method as described in Figure 7-6 . Use the standard normal distribution (Table A-2).
Hypothesis testingHypothesis testingDraw inferences about a populationbased on a sampleTesting a claim about a property ofa population
Statistical Inference∗ Inferences about a population are made onthe basis of results obtained from a sampledrawn from that population∗ Want to talk about the larger population fromwhich the subjects are drawn, not theparticular subjects!
What Do We Test ?∗ Effect or Difference we are interested in∗ Difference in Means or Proportions∗ Odds Ratio (OR)∗ Relative Risk (RR)∗ Correlation Coefficient∗ Clinically important difference∗ Smallest difference considered biologicallyor clinically relevant
Hypothesis TestingGoal: Make statement(s) regarding unknown populationparameter values based on sample data∗Elements of a hypothesis test:∗ Null hypothesis - Statement regarding the value(s) ofunknown parameter(s). Typically will imply no associationbetween explanatory and response variables in ourapplications (will always contain an equality)∗ Alternative hypothesis - Statement contradictory to the nullhypothesis (will always contain an inequality)
Null Hypothesis∗ Usually that there is no effect∗ Mean = 0∗ OR = 1∗ RR = 1∗ Correlation Coefficient = 0
Alternative Hypothesis∗ Contradicts the null∗ There is an effect∗ What you want to prove ?
Null Hypothesis expresses no differenceExample:H0: µ = 0Often said“H naught” Or any numberLater…….H0: µ1 = µ2
Alternative HypothesisH0: µ = 0; Null HypothesisHA: µ = 0; Alternative HypothesisResearcher’s predictions should bea priori, i.e. before looking at the data
Estimation: From the Sample∗ Point estimation∗ Mean∗ Median∗ Change in mean/median∗ Interval estimation∗ 95% Confidence interval∗ Variation
Parameters andReference DistributionsContinuous outcome data∗ Normal distribution: N( μ, σ2)∗ t distribution: tω (ω = degrees of freedom)∗Mean = (sample mean)∗Variance = s2(sample variance)Binary outcome data∗ Binomial distribution: B (n, p)X
Hypothesis TestingGoal: Make statement(s) regarding unknown populationparameter values based on sample dataElements of a hypothesis test:∗ Test statistic - Quantity based on sample data and nullhypothesis used to test between null and alternativehypotheses.∗ The test statistic is found by converting the sample statistic(proportion, mean or standard deviation) to