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Lecture 10 - The t Test for Two Independent Variables

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Alison Luby

PSY201H1F L10; Nov. 24, 2011 Assignment Info • Only need to go as far as text for assignment • Use APA style • Additional office hrs SS4040 5-7pm on Mon • Only paraphrasing last paragraph of intro and methods ○ When citing, put (assignment introduction) or (assignment methods) beside EACH piece of info (Glickstein, 2007) • When refer to study referenced in the given intro/methods: and-statistics-part-4.html ○ Ex. Lustig, Haser, and Tonev (in press) say … • How to know if significantly different pops, overlap (assignment introduction). • Like all hypothesis tests, general purpose of indep- • No pg limit, keep as short as possible measures t test: ○ Clear, concise, accurate ○ Determine whether sample mean dif obtained • No need for a running head but do include a title in research study indicates real mean dif btwn page the two pops (or treatments) or whether • No abstract obtained dif is simply result of sampling error • Report findings from math with APA style, use text and APA manual for style Assumptions? • No outliers Ch. 10: The t Test for Two Independent ○ Trim? Leave out? Resample? ○ Important to have no outliers so data not Variables Comparison of Two Samples skewed • Ex. Is there a dif in amt of aggressive bhvr in girls • Homogeneity of variance – F-Max test vs boys at age 11? • Similar sample sizes • What is effect of 0 mg vs 10 mg of particular drug ○ Otherwise pooled variance (larger n has larger on reducing seizures in rats? effect) • Which is more effective, cognitive bhvr therapy • A testable preconceived hypothesis… (CBT) or drug therapy for treating depression in ○ If want to test something hadn’t originally young teenagers? planned for, can, but ideally the original data • Concerns mean difs btwn 2 separate & indiv grps should become pilot data & need to grab a new sample to test this hypothesis Independent-Measures Design • Aka. Between-Subjects Design Resampling techniques (don’t need to know) • Expands single-sample t test to test for difs btwn two separate and independent grps • Evaluate mean difference between two pops or btwn two treatment conditions • Unknown population means (µ) & standard deviations (σ) ○ Estimate from sample data • Might not know anything about parent pop distrs but if samples representative then we know more • Normally pop variance, standard deviation, mean unknown Notation • Use subscripts to specify sample st • 1 sample: n ,1M ,1SS 1 • Resampling – sampling w replacement via • Second sample: n ,2M ,2SS 2 computer • Population mean: μ a1d μ 2 • H0: μ1-μ2= 0 (no mean difference) ○ Do since can’t apply t test to non-normal distr • H1: μ1-μ2≠0 (mean difference) • When all you sampled is all you can get, Hypothesis Testing w the Indep-Measures t expensive Statistic • Split sample into many subsamples Hypothesis Testing w the Indep-Measures t Statistic • Only dif btwn t formula and z-score formula: ○ z-score uses actual pop variance σ or standard dev ○ t formula uses the corresponding sample variance s (or standard dev) when pop value is not known ○ To calculate indep samples t statistic: • For independent t, technically a test of dif btwn Determine dif btwn sample means & estimate sample mean difference & hypothesized pop mean error term (estimated SEM) dif  2 sources of error contributing to standard ○ Independent t: error in denominator ○ To estimate standard error term (s (M1-M2) need to know how to calculate pooled variance or (H0: 1 -2 = 0) pooled standard dev ○ Single sample:  Pooled Variance = s = Wpighted avrg of variance in each sample distr; works for both = & ≠ size samples • Same logic behind hypothesis testing applied as w ○ Assume samples come from pops w equal one sample variances ○ Error term typically defined as a s/√n or s /√n, • However, important difs w two grps: ○ Incorporate 2 pop means into hypotheses so easy to see how to use the pooled variance ○ Look at hypothetical distrs of sample mean or pooled standard deviation to estimate SEM 1. Make decision difs when testing hypotheses ○ So: Estimated standard error of mean (SEM) ○ If t statistic ratio indicates obtained dif btwn includes 2 measures of variability & 2 sample sample means (numerator) is substantially sizes greater than dif expected by chance (denominator) reject H , conclude there is a ○ Degrees of freedom include the n of each 0 group real mean dif btwn the two pops or treatments  Ex. df = (n 11) + (n 21) = df +1df = 2 1 ○ Often don’t use 1-tailed test since looks suspicious to increase chance of reject H 0 + n 2 2 st • To prepare data for analysis: 1 step = compute ○ Most 1-tailed tests would be equally valid as a sample mean & SS (or s or s ) for each of the two 2-tailed test samples  If don’t get what looking for w 1-tail, have • Can still use skewed data and make seem more to retest, can’t use same data w a 2-tail normal Hypothesis test follows same 4-step procedure in Ch. 8, 9 1. State hypotheses, select α level ○ For indep-measures test, H 0tates that there is no dif btwn the two pop means 2. Locate critical region. Critical values for t statistic are obtained using df (add together df of each Plotting Your Results sample) ○ Ex. N1=19, N 220  df = (19-1)+(20-1)=37 3. Compute the test statistic. The t statistic for the indep-measures design has same structure as single sample t in Ch. 9. ○ However, in indep-measures, all components of t formula are doubled: 2 sample means, 2 population means, & 2 sources of error contributing to standard error in denominator • Ex 1: N = 20 (observations per grp); N = 1 2 1048576 (observations per grp) – Ridiculously Large Sample (overpowered expt, P = 0) ○ d1= 0.027394031, d = 2.015805989 2 2 -5 ○ r1 = 0.000192419, r 2= 6.2473x10
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