VIEWS: 3 PAGES: 31 POSTED ON: 10/5/2011
Hypothesis testing Behavioural Science II Week 1, Semester 2, 2002 Hypothesis testing • Null hypothesis is that there is no systematic relationship between independent variables (IVs) and dependent variables (DVs). • Research hypothesis is that any relationship observed in the data is real. Behavioural Science II 2 Hypothesis testing • Whereas research hypothesis tends to be imprecise about numerical differences between groups (e.g., difference in reaction times), null hypothesis states very specifically that difference should be zero. Behavioural Science II 3 Null hypothesis versus alternative hypothesis • The null hypothesis assumes that scores for different levels of the IV are random samples from the same population. • The alternative hypothesis is that samples come from different populations. Behavioural Science II 4 Null hypothesis versus alternative hypothesis • For any single experiment, we are bound to see a difference, just as we see a difference between the means of two random samples in a distribution of sample means. • If the null hypothesis is true, then differences in mean scores are just two random samples from the same population. Behavioural Science II 5 Testing the null hypothesis • A statistical test assesses the probability of obtaining a given sample or samples of scores, assuming the null hypothesis is correct. Behavioural Science II 6 Testing the null hypothesis • If the probability is low enough (e.g., p<.05), then the null hypothesis is rejected in favour of the alternative (research) hypothesis, and the IV is deemed to have a systematic effect. • If the probability is not sufficiently low (e.g., p>.05), then the null hypothesis is not rejected but retained, and the IV is deemed to have no effect (i.e., the observed changes are due to chance). Behavioural Science II 7 Statistical significance • Statistical significance refers to the probability of the data obtained, given that the null hypothesis is true. • A statistically significant result does not mean that the null hypothesis is improbable. • There is an ongoing gap between statistical significance and substantive significance. Behavioural Science II 8 Hypothesis testing and sampling distributions • The decision to reject or not reject the null hypothesis usually is made with reference to the sampling distribution of a statistic of some kind (e.g., z-distribution, t- distribution). Behavioural Science II 9 Example of hypothesis testing using z-distribution • Null hypothesis population parameters: = 15 =15 • Random sample statistics Mean = 110 N=9 Behavioural Science II 10 Applying formulae 15 15 X 5 N 9 3 X X 110 100 10 Z 2 X 5 5 • Given that z-score of 1.96 = p< .05 (two- tailed), would reject null hypothesis. Behavioural Science II 11 Example of hypothesis testing using t-distribution • Null hypothesis population parameters: =100 • Random sample statistics Mean = 110 N=9 ∑x2 = 960 Behavioural Science II 12 Applying formulae Given that t- scores of ˜ x 2 960 960 10.95 2.306 (df=8) N 1 9 1 8 =p< .05 (two-tailed), would reject the ˜ 10.95 10.95 ˜X 3.65 null N 9 3 hypothesis. X X 110 100 10 t 2.74 ˜X 3.65 3.65 Behavioural Science II 13 Hypothesis testing using confidence intervals • We reject null hypothesis when null population mean lies outside the confidence interval. • We infer alternative population mean is higher than null population mean if lower limit of confidence intervals is to right of null population mean and lower if upper limit of confidence intervals is to left of null population mean. Behavioural Science II 14 Errors in hypothesis testing • Given the gap between statistical and substantive significance, a decision based on probability to retain or reject the null hypothesis can be wrong. Behavioural Science II 15 When null hypothesis is true (Type I error) • When null hypothesis is true, and it is rejected, this decision is called a Type 1 error. • The probability of making such an error is designated alpha () and is equivalent to the significance level (e.g., p<.05). Behavioural Science II 16 When null hypothesis is true (Type I error) • If null hypothesis is true and alpha level is set at .05, then the null hypothesis will be rejected 5% of time even though it is true. • One way to safeguard against a Type I error is to set a more stringent alpha level (e.g., p<.01). Behavioural Science II 17 When null hypothesis is false (Type II or III errors) • When alternative hypothesis is true, and the statistic (mean) from alternative distribution falls within cut-off points (i.e., p>.05), then null hypothesis would be retained. Behavioural Science II 18 Type II error • Retaining null hypothesis when alternative hypothesis is true is called a Type II error. • The probability of making a Type II error usually is symbolized as beta (). • The probability of beta depends on how much the alternative hypothesis sampling distribution overlaps the retention region of the null hypothesis sampling distribution. Behavioural Science II 19 Type III error • It is also possible to make a Type III error, by rejecting a null hypothesis but inferring the incorrect alternative hypothesis. • The probability of making a Type III error usually is symbolized as gamma () and is equivalent to whatever percentage of scores in the alternative distribution falls in the far end of the null hypothesis distribution. The probability of making a Type III error is usually quite small. Behavioural Science II 20 The power of a test • The probability of rejecting a false null hypothesis and correctly inferring the position or direction of the alternative hypothesis with respect to the null hypothesis. • Factors affecting power and error rates Behavioural Science II 21 Power is affected by significance (alpha) level • Setting a less stringent significance level increases the discriminatory power of the statistical test and increases power as long as the alternative hypothesis is true. Behavioural Science II 22 Power is affected by magnitude of difference between sample means • So, increasing the difference in the size of the mean at differing levels of the IV increases the power of the test. Behavioural Science II 23 Power is affected by sample size • An increase in sample size increases the power of the test, if the alternative hypothesis is true. • This is because as sample size increases, the standard error of the mean decreases, thus reducing the overlap between the null and alternative hypotheses. Behavioural Science II 24 Effect size • In order to gauge the effect of the IV, it makes sense to contrast the difference between the population mean for the null hypothesis and the population mean for the alternative hypothesis. Behavioural Science II 25 Effect size formula 0 1 Eff ect_ size • where • is standard deviation of population of dependent measure scores. Behavioural Science II 26 Judging effect sizes • According to Cohen (1988) .20 = small effect size .50 = medium effect size .80 = large effect size Behavioural Science II 27 Do we really need the null hypothesis? • A significant test of the null hypothesis does not mean the data are not a product of chance. • The significant result may simply be a Type I error (falsely rejecting null hypothesis). Behavioural Science II 28 Do we really need the null hypothesis? • Better to test research hypothesis, if know size and direction of effect. • Even better report combination of outcome values (e.g., effect sizes, confidence intervals, strength of relationship). Behavioural Science II 29 One-tailed versus two-tailed tests • Conventionally reject null hypothesis if obtained z-score or t-score falls beyond certain values in either tail of the relevant sampling distribution (i.e., a two-tailed test). • In specific contexts, a one-tailed test might seem appropriate (e.g., reject null hypothesis only if test statistic fell in 5% left-hand tail of distribution. Behavioural Science II 30 One-tailed versus two-tailed tests • Generally, two-tailed tests are preferred to one-tailed tests. • The IV may have an effect in opposite direction to the one predicted. Behavioural Science II 31