Session 3 - The Special Case of Two Groups The t Test.docx

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University of Toronto St. George
Cell and Systems Biology
William Navarre

HMB325H © Lisa | Page 13 L E C T U R E 3 : T H E S P E C I A L C A S E O F T W O G R O U P S : T H E t T E S T ( C H A P T E R 4 ) LEARNING OBJECTIVES 1. discuss the process for hypothesis testing (null) & describe how a study result & the standard error are used to determine the test statistic & the resulting probability (P) value 2. define alpha (α) level & explain how it is used to determine whether a given study result is statistically significant 3. discuss the effects of multiple significance testing & how these should be addressed 4. describe & carry out the processes to complete a t test 5. calculate the confidence interval for the difference bw means 6. describe the rational for the t test 7. describe the assumptions of the t test & confirm that each has been met PROBABILITY (P) VALUE • one or more of the following explain a study’s observed results: 1. the observed effects are real (but we can only assess this indirectly) 2. due to bias in study design or conduct (assess by critically appraising the study) 3. due to chance (assess using the applicable statistics) • ex. study of the efficacy of diuretics (drugs that increase urine production)  3 groups of 24 indiv are studied to compare a new diuretic, the standard drug, & placebo  independent variable: exposure measure  indicator variable: group (nominal variable)  dependent variable (outcome of interest): urine output (mL/day; continuous variable)  result was consistent w what was expected  new drug was the most effective diuretic  even if all 3 were ineffective (like placebos), the outcomes wouldn’t be the same due to random variability (need to use a statistical test ANOVA to assess) • we need to assess the probability of results as ‘extreme’ as those observed occurring by chance  use ANOVA (F-ratio is the ‘test statistic’) • if we conclude that the result is not due to chance, but due to a ‘true’ effect (determined by ANOVA), could we be wrong?  what is the probability that we will be wrong? • P: the risk of getting a false positive error ALPHA (α) LEVEL • how statistically significant? how low does the value have to be to set aside chance to be a reasonable effect? • traditionally, the probability of observing results as ‘extreme’ as those observed must be less than 5% (1/20, P < .05) to conclude it is a ‘statistically significant’ result  this cut-off value is called the alpha (α) value (it is expected to be chosen BEFORE a study is started) • α is the error rate • what does the chosen α pre-determine? • analysis of variance (ANOVA) only indicates whether the group means differ from each other more than expected by chance  it doesn’t indicate which group(s) differ from the rest • this requires the use of additional, follow-up tests (ex. t tests) to carry out pair-wise group comparisons THE t TEST • t test: a special case of ANOVA where only 2 groups are being compared • assesses the size of the difference bw the 2 means (we can now simplify the calculations) 2  between group variance (SD bet) is just the difference bw the 2 group means • if there truly is no effect (null hypothesis), what is the ‘expected’ difference bw the 2 group means? (= chance is the only reason for differences bw the means)  the expected difference bw the 2 group means should be 0 • what is the size of the observed group difference compared to?  need to compare the amount of variability (difference bw the 2 group means) that is already inherent in the data to the standard error  shows how much variability is inherent in the data 1) the more variable the data is, the more possible it is that the 2 independent group means differ from each other 2) little variability inherent in the data – the 2 independent group means should be very close • the ratio of the difference bw the observed means & the standard error of the difference (SED): difference mean −mean t= = 1 2 SED 2 2 SD 1+ SD 2 √ n n 1 2 SED: how big of an effect we expect to observe simply by chance alone (by the inherent variability) SD : variance n: sample size of each group • in the instance where the 2 group variances (*&sample #) are similar to one another (may be just 2 independent estimates of the same amount of variability in the data), a stronger technique is used • ‘pooled’ variance: usually a better estimate than using the 2 indiv group variances 2 2 2 (n1–1 S) +(1 – 1)2D 2 SD pooled n1+n 22 (n – 1): degrees of freedom for one variance denominator = total degrees of freedom • pooled variance: avg variance bw the 2 variances rather than using each group variance t TEST MISUSE • the t test is commonly misused to carry out multiple pair-wise comparisons • instead of using ANOVA to test multiple groups at once • take the group that has the highest mean & the one w the lowest mean & conduct a t test • if that was statistically significant, they would take the next pair of means (next largest), & so on until they stop getting statistically significant results • the challenge is that for each one of these tests, they would use α of .05 • if you are doing multiple tests, α of .05 = 1/20 that the observed result may have risen simply by chance • if we are now doing 3 tests (3 groups), how many pair-wise comparisons can we make?  compare new drug to placebo, standard drug to placebo, new drug to standard drug (3 pair-wise comparisons) • if
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