Chapter 15 Comparing Two Populations Dependent samples Dependent
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Chapter 15 Comparing Two Populations: Dependent samples
Dependent Samples l Dependent Samples - measurements are somehow related to one another either because: – The measurements came from the same subject (Within-subjects design), or – Subjects were paired or matched (Matched-pairs design) l May also be called a “Repeatedmeasures” designs
Why use dependent samples? l Advantages – Fewer subjects required with a withinsubjects design (each subject is measured twice) – Increased power through decreased variability l Disadvantages – “Carry-over” effects
Carry-over Effects Let’s say we are interested in methods to teach reading to pre-schoolers l We l – introduce the whole-language method, measure, – then use the phonics method, and measure l We have a repeated-measures (withinsubjects design), but
Carry-over Effects How can we remove the effects of teaching reading with the wholelanguage method? l There is a “carry-over” to the next condition (phonics) condition, because the preschoolers may have learned to read l Do we expect to forget how to read once they learn to? l
Carry-over Effects Matched-pairs designs will eliminate the carry-over effect problem l Matching, though, must be done well, without respect to the obtained scores, l – matching is done before the experiment l Along a dimension that is relevant to the study
Dependent Samples t-test With pairs of scores, either the result of matching or taking multiple measurements from the same subject l Calculate the difference (D) between the two scores (maintaining positive and negative values) l Calculate the mean difference, standard error, and then t l
Dependent Samples t-test
Dependent Samples t-test MD is the mean of the differences s. D is the standard deviation of the differences s. MD is the estimated standard error of the sampling distribution of MD Δ 0 is the difference between the population means (parameter)
Example of a Dependent Samples t-test We want to compare a new teaching method with a traditional method on math l 2 groups - 10 12 th grades in each, l Group “Old” receives old method, Group “New” gets new method l Subjects are matched on their IQ l
Hypothesis test of Old vs. New (teaching math) l 1. State and Check Assumptions – About the population l l Normally distributed? - don’t know Homogeneity of variance – we’ll check – About the sample l l Independent Random sample? – yes Dependent samples – About the sample l Interval level
Hypothesis test of Old vs. New (teaching math) l 2. Hypotheses HO : μOld = μNew (the effectiveness of the old and new method are the same) μOld - μNew = 0 (the difference between the effectiveness of Old and New is 0) HA : μ 1 ≠ μ 2 (the effectiveness of Old and New are not equal) μ 1 - μ 2 ≠ 0 (there is a difference between the effectiveness the Old and New)
Hypothesis test of Old vs. New (teaching math) l 3. Choose test statistic – 2 groups l dependent samples Dependent-sample t-test
Hypothesis test of Old vs. New (teaching math) l 4. Set Significance Level α =. 05 Critical Value Non-directional Hypothesis with df = np - 1 = 10 - 1 = 9 From Table C, critical value of α /2 =. 025 tcrit = 2. 262, so we reject HO if t ≤ - 2. 262 or t ≥ 2. 262
Hypothesis test of Old vs. New (teaching math) l 5. Compute Statistic – We need:
Scores on the WAT (Wisconsin Achievement test) Old 78 55 95 57 60 80 50 83 90 70 New 74 45 88 65 64 75 41 68 80 64 D 4 10 7 -8 -4 5 9 15 10 6 Need: Δ MD s. MD
Scores on the WAT Old 78 55 95 57 60 80 50 83 90 70 New 74 45 88 65 64 75 41 68 80 64 D 4 10 7 -8 -4 5 9 15 10 6 ΣD = 54 ΣD 2 = 712 MD = 5. 4 SS(D) = 420. 4 s 2 D = 46. 7111 s. D = 6. 833 np = 10 s. MD = s. D/√np = 6. 833/ √ 10 = 2. 161
Example (non- directional HA) t is greater than tcrit, reject HO
Hypothesis test of Old vs. New (teaching math) l 6. Draw Conclusions – because our t falls within the rejection region, we reject the HO, and – conclude that the old and new method differ in their effectiveness in teaching math, with the old method superior
Comparing Dependent and Independent Samples t-tests l Suppose, instead of pairing subjects based on IQ, we compared the mean of the two groups using a 2 independent samples t-test
Scores on the WAT l Old 78 55 95 57 2219. 6 60 80 50 83 90 70 l ΣXO = 718 ΣX 2 O = 53772 MO = 71. 8 SS(XO) = s 2 O = 246. 622 s. O = 15. 70 n. O = 10 New 74 45 88 65 1902. 4 64 75 41 68 80 64 ΣXN = 664 ΣX 2 N = 45992 MN = 66. 4 SS(XN) = s 2 N = 211. 377 s. N = 14. 54 n. N = 10
Independent Samples t-test Cannot reject HO
Comparison (cont. ) By matching subjects, a priori (prior to the experiment) l the variance is reduced substantially, thereby l Allowing us to find a difference that was obscured l
Ranking When completing a Wilcoxon, ranking must be done with the smallest difference receiving the lowest rank (1) l The biggest difference gets a rank of np l
Hypothesis Test with nonparametric alternative to t-test Nicotine patches supposedly reduces smoking l A psychologist records the number of cigarettes each of 9 patients smokes per day without a patch and with a nicotine patch. l
OOPS!
Hypothesis Test l 1. State and Check assumptions – Random Sample - yes – Dependent Sample (Within-subjects design) l measures come from the same person (wearing the patch, and not wearing the patch) – Number of cigarettes smokes normally distributed? - not sure – Homogeneity of Variance - nope
Hypothesis Test l 2. State Hypotheses HO : μBEFORE = μAFTER HA : μBEFORE > μAFTER
Hypothesis Test l 3. Choose Test – 2 dependent samples – H of V - no Wilcoxon Tm (Wilcoxon signed-ranks test)
Hypothesis Test l 4. Set Significance level α =. 05 with np > 8, the Wilcoxon Tm approaches a z directional alternative at. 05, using Table A z = 1. 65 if obtained z ≥ 1. 65, Reject HO
Hypothesis Test l 5. Compute Test Statistic
Wilcoxon Test l Non-parametric alternative to dependent sample t-test – Calculate the differences between paired scores (D) – Calculate the absolute value of each (|D|) – Rank |D| – “Sign” the ranks using the sign from D – Add all positive Ranks (T+)
Hypothesis Test l 6. Draw conclusions – Since the obtained z was greater that the critical z of +1. 65, HO is rejected, and – Conclude that the distributions are different with the before values greater than the after values
Review Procedure for calculating t Independent samples l Compute (for each group): l
Calculating t - Independent Samples l To find t, the estimated standard error of the sampling distribution is calculated:
Computing t l The t statistic ( a family of distributions which varies with df) is computed using:
Review Procedure for finding t l Dependent Samples – matched-pairs, correlated samples, repeated-measures, etc. (2 samples) – Compute D (the difference between pairs of scores) l np (the number of pairs of scores l MD (the mean of the differences) l
Dependent Samples t (cont. ) – Compute s. D (the standard deviation of the differences) l s. M (the estimated standard error) D l
Estimated Standard Error and t
Conceptually, what is the t statistic? In both tests, the numerator is a measure of the observed difference between scores (either unpaired or paired) l The denominator, of both tests, is an estimate of the standard deviation of the sampling distribution of M 1 - M 2 (we call it “standard error”) l
Error Terms When a sample is taken from a population, statistics computed from the sample are estimates of parameters, however, l A certain amount of variability is expected (no two sample means are exactly alike) l This variability from sample to sample is called “error” l
Error and t-tests The amount of error obtained from a sample is used to estimate the “standard” error that might be expected in the situation (given n, etc. ) l This error can produce large (significant) differences between sample statistics, just by chance, because no two sample means are exactly alike l
t statistic as a ratio l Therefore, obtained difference t = ———————— difference expected by chance (“error”) We will return to this type of conception again
Can’t tell the difference between independent and dependent samples? If measurements are related to one another - dependent samples l You will not be told when to use independent or dependent samples ttests l You have to decide l
Example A psychologist believes that environment is more important than genetics in influencing intelligence. l She locates 12 pairs of identical twins that have been reared apart, one twin in each pair in an enriched environment and the other in an impoverished one. l Independent or Dependent Samples? l
Another A new experimental drug, ABZ, is thought to have beneficial effects on AIDS. l 20 AIDS patients are randomly selected and assigned to one of two conditions: ABZ for 90 days or Placebo for 90 days. l Independent or Dependent Samples? l
A third example A researcher suspects that increasing the level of lighting during the winter months will increase mood. l The researcher selects 36 students, tests their mood, then replaces the light bulbs in the dorm from 75 -W to 100 -W for 1 month, then tests each student again l Independent or Dependent Sample? l
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