PSYS 054, Statistics for Psychological Science 2/23/16
PSYS 054, Statistics for Psychological Science 2/23/16 PSYS 054
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This 7 page Class Notes was uploaded by Delaney Row on Friday February 26, 2016. The Class Notes belongs to PSYS 054 at University of Vermont taught by Keith Burt in Fall 2016. Since its upload, it has received 36 views. For similar materials see Statistics for Psychological Science in Psychlogy at University of Vermont.
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Date Created: 02/26/16
PSYS 054 – Statistics For Psychological Science Notes for the week of 2/22/16 – 2/26/16 2/23/16 Paired t-‐test: Overview • A crucial point to remember is that we are using paired t-‐tests when we are using a pair of “related” scores • Examples: o Repeated measures on the same person (having a pre-‐test and a post-‐ test) o Ratings from pairs of spouses or family members o Longitudinal data Paired t-‐test: Assumptions • Paired t-‐test loses the assumption of independence of observations • With the paired t-‐test, we are observing difference scores o This is: one score (#1) minus the other (#2) (also can be thought of pre-‐test minus post-‐test) • Paired t-‐test assumed that the distribution of the difference scores is relatively normal Difference Scores • Notation: with the pre-‐test of a pair being Xsub1 and the post-‐test being Xsub2… o Then, D = (Xsub1 – Xsub2) = the difference score • Once we get the difference score, we are basically following all the steps we do in a one-‐sample t-‐test, with the difference score being the main data we are looking at o Degrees of freedom for paired t-‐tests will always be: (Number of difference scores -‐1) o Use the M and SD of difference scores in the formula Paired t-‐test: Hypothesis specification Symbol Interpretation µ 1 Population mean of score set #1 µ Population mean of score set #2 2 μ Population mean of difference scores D • In all cases… Dμ 1 µ2 -‐ µ • For a two tailed test… o H : 0 D = 0 § This is always going to be the null hypothesis for any paired t-‐ test (it’s saying that there is no difference in scores) o H : 1 D ≠ 0 o Example from class: § “A stats professor is interested in whether or not students’ scores on Exam #2 will be higher or lower than scores on Exam #1. The professor has no specific prediction about which exam scores will be higher or lower.” • For a one tailed test in the positive direction… o H : 0 D = 0 § Notice the null hypothesis stays the same o H : 1 D > 0 § For a one tailed test, the alternative hypothesis must have the mean difference either be great or less than zero, depending on its direction o Example from class: § “35 individuals diagnosed with Major Depressive Disorder (“clinical depression”) are given a baseline questionnaire of depressive symptoms at the first visit of a clinical trial for a new medication. They complete the questionnaire again after taking the medication for 6 weeks. The researchers hypothesize that the medication will reduce depressive symptoms over time.” • For a one tailed test in the negative direction… o H : 0 D = 0 o H : 1 D < 0 o Example from class: § “A researcher is interested in studying the effect of age on perceived conflict in romantic relationships. She gives each partner from 15 couples a questionnaire assessing their reports of conflict in the relationship. The scores are set up so that younger partners are set #1 and older partners are set #2. The researcher has reason to predict that older partners will report more conflict.” Paired t-‐test: Formula and Notation • What you need: o Mean of difference scores (Dbar) o SD of difference scores (S sub D) o Standard error of difference scores (SDbar) o Number of difference scores (N sub D) Paired t-‐test: Example • Baby sign language research Paired t-‐test: Sample Data • Sample of 8 infants assessed two times (at 15 months and 24 months) o 15 months = x1 (or pre-‐test) and 24 months = x2 (pre-‐test) • Researchers are predicting an increase over time, so it will be a one tailed negative test o Since it’s x1-‐x2 and x2 is suspected to be bigger, it will be one tailed negative Paired t-‐test: Calculations • Hypotheses o H : 0 μD = 0 o H : 1 D μ < 0 • Calculations o D 1 = 6-‐9 = 2 -‐3; D = 2-‐7 = -‐5… • 2 D Dbar D – (D – Dbar) Baby 15 24 Dbar ID Months months D -‐3 -‐2.75 -‐0.25 0.06 1 6 9 -3 -‐5 -‐2.75 -‐2.25 5.06 2 2 7 -5 0 -‐2.75 2.75 7.56 3 5 5 0 -‐7 -‐2.75 -‐4.25 18.06 4 1 8 -7 5 4 8 -4 -‐4 -‐2.75 -‐1.25 1.56 6 8 10 -2 -‐2 -‐2.75 0.75 0.56 7 9 6 3 3 -‐2.75 5.75 33.06 8 3 7 -4 -‐4 -‐2.75 -‐1.25 1.56 2 - ∑(D-‐Dbar) Mean of difference scores = 2.75 = 67.48 SD of difference scores = 3.10 • D for the most part is negative (what researchers expected) • We are replacing the two sets of scores with the difference score o We are finding M and SD of difference scores, not of the other sets of scores Paired t-‐test: Calculations (continued) • Dbar = mean of difference scores = -‐2.75 • s D SD of difference scores = 3.10 • Paired t-‐test formula: o t = Dbar / D (s /D √N ) o = -‐2.75 / (3.10 / √8) o = -‐2.75 / (3.10 / 2.83) o = -‐2.75 / 1.10 o = -‐2.50 (df = 8-‐1 = 7) • Degrees of freedom = 7 • Note: the denominator in the formula is already accounting for standard error Paired t-‐test Example: Decision • t = -‐2.50 • One-‐tailed test, negative direction o (α = .05) o df = 8-‐1 = 7 • From Appendix E.6, critical value = -‐1.895 • -‐2.50 < -‐1.895, therefore: reject the null hypothesis Reporting t-‐test: APA Style • Italicize notation letters, but not numbers • Include degrees of freedom in parentheses • If calculated t falls in rejection region, write “p < .05” or report exact p if you are using SPSS o Or write “n.s.” (meaning ‘not significant’), e.g. “ t(11) = 0.53, n.s.” • So for our paired t-‐test example: o t(7) = -‐2.50, p < .05 o NOTE: lower case t should be italicized and then the 7 in parentheses Independent Samples t-‐test • The most common t-‐test used in actual research o We will most likely be using this test on our final projects! • This test compares independent sample means o For example, the difference between mean of variables X an1 X , 2 when these represent scores from different, unrelated individuals One-‐sample t-‐test Paired t-‐test Independent samples t -‐test Compares ONE sample Compares TWO sample means Compares TWO sample means, mean to a hypothesized that are related (through even if they are unrelated population mean difference scores) Independent Samples t: Hypothesis • In the population, which group has the larger mean? • Similarly to a paired t-‐test, hypotheses are written in the form of population mean differences o H : 0 1µ 2-‐ µ = 0 (null hypothesis) § This is saying that the mean scores from each group will be equal/have no difference o H : 1 1µ 2-‐ µ ≠ 0 (two-‐tailed research hypothesis) o H : 1 1µ 2-‐ µ < 0 (one-‐tailed research hypothesis) …OR… o H : µ -‐ µ > 0 (one-‐tailed research hypothesis) 1 1 2 • t is calculated in a similar fashion as well, BUT: • The standard error calculations are somewhat more complex • Degrees of freedom are calculated differently Real-‐Life Research Using Independent Sample t-‐test • Examples from class: o Fingerson (2008), studying psychopathy and trauma: “…an independent-‐samples t-‐test found no significant differences between psychopaths and non-‐psychopaths with regard to their exposure to trauma… ” o Gibson (2008), studying ethnicity (Caucasian vs. African-‐American) and exposure to thin-‐ideal images in female college athletes: “…an independent samples t-‐test was conducted… the test was significant; white female athletes have higher exposure to magazines than non-‐ white female athletes… t(79) = -‐2.44, p < .01…” Formula for Independent Samples t • The denominator represents another type of standard error; the variances of the group • Degrees of freedom is different here: df 1=(N 2 + N -‐ 2) • NOTE: memorize the different degrees of freedom formulas; they will not be on the formula sheet for exams! Independent Samples t-‐test: Assumptions • Independent t-‐test assumes: o (1) Data are independently sampled… o (2) From a normally distributed population… § Normality assumption o (3) Variance of the two groups is the same in the population § “Homogeneity of variance assumption” Independent Samples t-‐test: Example • Researchers want to know if learning baby signs impacts language development (positively or negatively) by age 2 o This will be a two tailed test because there is no specific direction the researchers are going in • Group A: asked to pay special attention to baby’s language • Group B: taught a series of baby signs • Outcome: standardized language assessment score (1-‐5 range) • N = 10 infants per group • Results: o Group A: M = 3.20, SD = 1.79 o Group B: M = 4.00, SD = 2.61 Thinking About Populations • μ 1 = mean of whatever population Group A is drawn from • μ 2 = mean of whatever population Group B is drawn from • We want to know: could samples have come from populations with the same mean? o If true, difference in group means is due only to sampling error • H :0 μ1 2‐ μ = 0 • H :1 μ1 2 μ ≠ 0 o This alternative hypothesis shows it’s a two tailed test Independent t-‐test: Applying Steps 1. Info: M and SD from two independent samples 2. Appropriate test: independent samples t-‐test! 2 2 3. t = 1X bar -2‐ X bar /s1r(s /1 +2 s2/N ) 4. df = (1 +2 N -‐ 2) = 10 + 10 -‐ 2 = 18 5. Two-‐tailed test, .05 alpha level, degrees of freedom: use Appendix E.6 6. Compare calculated t statistic to critical values • Note: a two-‐tailed test has two critical values, positive and negative! • If calculated t is greater than positive or less than negative critical value, reject the null hypothesis, otherwise retain Calculating t for the Example • This can be tricky; we are given SD but we need variance, so we need to square each SD to get the variances of each group • t = (3.20 -‐ 4.00) / sqr(3.20/10 + 6.81/10) = (-‐.80) / sqr(.320 + .681) = -‐.8 / sqr(1.001) = -‐.8 / 1.00 = -‐.8 • Or since t’s are written with one or more numbers before the decimal and two after, t(18) = -‐0.80. Appendix E.6 for Example: Two-‐Tailed • Critical values of t for a two-‐tailed test, alpha of .05, df=18, are: ± 2.101 • -‐0.80 is not more extreme than -‐2.10, therefore: we will retain the null hypothesis o There is not enough information to suggest that these two samples come from different populations…
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