# Chi-Square P216; 269

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1 Chi-Square P16; 69

2 Confidence intervals CI: % confident that interval contains population mean (µ) % is determined by researcher (e.g. 85, 90, 95%) Formula for z-test and t-test: CI = M +/- z*(σ M ) CI = M +/- t*(s M ) Where z* or t* is the critical value Example: CI = 86 +/- 1.96(1.7) = 8.67 to % confident pop mean in this range

3 Inference by eye: Interpreting CI s Why use CI instead of null hypothesis? Move beyond dichotomous decision Write-up suggestions We are 90% confident that the true improvement on the test ranges from to Values outside of this range are relatively implausible. The lower limit of is a likely lower bound estimate of improvement. Rules of eye What do bars on a graph represent (i.e. sd, se, CI)? Focus on the means, but take variability into account Interpret CI any value outside CI is unlikely Double SE bars to get approximate 95% CI

4 Chi-square Kristen is interested in evaluating whether the method of cooking potato chips affects the taste of the chips. She tests 48 individuals. Each tastes 3 types of chips, then select which chip they prefer to eat. Method of cooking potato chi ps Fried in animal f at Fried in Canola oil Baked Total Observ ed N Expected N Residual Chi-Square a df Asy mptotic Signif icance Test Statistics Number who Method of pref erred cooking each type of pot ato chips chip a. 0 Cells.0% low f reqs 16.0 expected low...

5 Nonparametric tests Parametric tests: assumptions about population distribution t-test, ANOVA Compare means & standard deviations Nonparametric tests Chi-square Compare frequency counts Evaluate hypotheses about population frequencies No assumptions regarding population distribution (M, s)

6 Chi-Square (X ) Goodness of fit test Uses proportions from sample to examine hypothesis about population proportions One sample, one variable (but, as many levels as needed) Test of independence Examines if there is a relationship between two variables One sample, two variables Similar to correlation with discrete variables

7 Chi-square (X ) Frequency table of data Observed frequencies (f o ) Compare to null hypothesis Expected frequencies (f e ) Null hypothesis No preference No difference vs. comparison population Expected frequency f e = pn ( f o f x Chi-square equation e ) df = C 1, where C = # of categories Chi-square distribution: Table A.4 p410 f e

8 Chi-square test Compare observed (O) and expected counts (E) Measure distance If distance = 0 then no difference between O & E = null hypothesis If distance > 0 then difference between O & E = evidence against null hypothesis Chi-square distributions df based on #cells not n

9 Goodness of fit: 1 sample Art appreciation: Painting shown to 50 participants. Asked to hang picture in correct orientation. 4 possible ways can hang painting H 0 : no preference (5% chance/orientation) Expected frequencies:.5(50) = 1.5 Top up (correct) Bottom up Left up Right up Observed frequencies Top up (correct) Bottom up Left up Right up

10 Chi-square table What is critical value? X cv α =.05 df = 4 1 =

11 Calculate Chi-square Observed Frequencies Expected Frequencies X ( f o f e ) f e X (18 1.5) 1.5 (17 1.5) 1.5 (7 1.5) 1.5 (8 1.5) Exceeds critical value (7.815), so reject H o

12 Chi-square write-up A chi-square statistic was calculated to examine if there is a preference among four orientations to hang an abstract painting. The test was found to be statistically significant, X (3, n = 50) = 8.08, p<.05. The results suggest that participants did not just randomly hang the art on any orientation. Instead it appears the correct top-up orientation (p = 18/50) and bottom-up orientation (p = 17/50) were used more often than the left side-up (p = 7/50) or right side-up (p = 8/50) orientation.

13 Goodness of fit: Chi-square Assess whether women become less depressed, remain unchanged, or become more depressed after giving birth Tested 60 women pre and post-birth Null hypothesis: no difference Postpartum Depression less depress ed same more depressed Total Observ ed N Expected N Residual

14 SPSS: Chi-square less depressed same more depressed Postpartum Depression

15 SPSS: Chi-square Postpartum Depression Test Statistics less depressed same more depressed Total Observ ed N Expected N Residual Postpartum Depression Chi-Square a df Asy mptotic Signif icance.00 a. 0 Cells.0% low freqs 0.0 expected low... Postpartum Depression Test Statistics less depress ed more depressed Total Observ ed N Expected N Residual Postpartum Depression Chi-Square a.037 df 1 Asy mptotic Signif icance.847 a. 0 Cells.0% low freqs 13.5 expected low...

16 Write-up A one-sample chi-square test was conducted to assess whether women become less depressed, remain unchanged, or become more depressed after giving birth. The results were found to be significant, X (, n = 60) = 1.70, p <.01. The proportion of women who were unchanged (55%) was greater than the hypothesized proportion (33%). While women who became less depressed (3%) and more depressed (%) were approximately the same to the hypothesized proportion. A follow-up test indicated that the proportion of women who became less depressed did not significantly differ from women who became more depressed, X (1, n = 7) = 0.04, p = ns. Overall, these results suggest that women in general do not become more or less depressed after childbirth.

17 Goodness of fit hypotheses Compare proportions of cases with hypothesized values Are there more smokers in study than nonsmokers? No difference in categories H 0 : 50% smokers; 50% non-smokers Or specific proportions obtained from prior studies H 0 : 0% smokers; 80% non-smokers

18 Write-up A chi-square goodness of fit test indicates there was no significant difference in the proportion of smokers identified in the current sample (19.5%) as compared with the value in the nationwide study (0%), X (1, n = 436) =.07, p =.79.

19 Assumptions for Chi-square Independence of observations Each subject gives 1 response Each observation is a different subject Size of expected frequencies Should not be less than 5 per cell Avoid by using larger samples Distorted when f e is very small

20 Chi-square (X ) Goodness of fit test Null hypothesis No preference hypothesis Test for independence variables! Null hypothesis Same proportions for variables i.e. no relationship between variables

21 Treating Cocaine Addiction Desipramine Lithium Placebo Success Relapse

22 Independence Chi-square test: variables Examine relationship between variables Treating cocaine addiction Relapse No Yes Total Desipramine Lithium 6 18 Placebo 4 0 Total 4 48 Compare study findings to expected findings f e f c n f r No relapse: 4*4/7 = 8 (successes expected/drug) Yes relapse: 48*4/7 = 16 (failures expected/drug)

23 Independence Chi-square test X Null hypothesis: no relationship between type of drug and relapse df = (R 1)(C 1) = -1(3-1) = Critical value = 5.99 X (14 8) 8 Significant! ( f o f e ) (6 8) 8 f e (4 8) 8 (10 16) 16 Examine proportions (#/total) of no relapse 10.5 Drug1: 14/4 =.58, Drug: 6/4 =.5, Drug3: 4/4 =.17 (vs. expected: 8/4 =.33) (18 16) 16 Relapse No Yes Desipramine 14 / 8 10 / 16 Lithium 6 / 8 18 / 16 Placebo 4 / 8 0 / 16 (0 16) 16

24 Write-up A chi-square test was conducted to assess whether cocaine addicts would relapse when treating addiction with Desipramine, Lithium, or a placebo drug. The results were found to be significant, X (, n = 7) = 10.5, p <.05. The proportion of addicts who did not relapse when treated with Desipramine (58%) was greater than the hypothesized proportion (33%). While the number of addicts who did not relapse when treated with Lithium (5%) and the placebo (17%) were approximately the same than the hypothesized proportion. The results suggest that there is a relationship between probability of a relapse and drug used to treat the addiction. It appears that Desipramine assists cocaine addicts to prevent relapse of drug abuse.

25 Gender and smoking example Is there an association between gender and smoking behavior? Is the proportion of males that smoke the same as the proportion of females?

26 Write-up A chi-square test for independence indicated no significant difference in the proportion of males or females that smoke, X (1, n = 436) = 0.494, p =.48.

27 Other non-parametric statistics Mann-Whitney U test Equivalent to independent-samples t-test DV (scores) converted to ranks; examine median Ex. IV: gender; DV: self-esteem rank Wilcoxon signed rank test Equivalent to paired-samples t-test Converts scores to ranks; compares T1 & T Kruskal-Wallis test Equivalent to one-way ANOVA Scores converted to ranks; 1 bet-ss IV w/ 3+ levels Friedman test Equivalent to repeated-measures ANOVA Scores converted to ranks; 1 w/in Ss IV w/ 3+ levels

28 Chapter 13 Quasi-experimental research Non-manipulated IV: Ss not randomly assigned to condition Lacks internal validity (confounding variable?) Quasi-experimental designs One sample Single-group posttest only design Single-group pre/posttest design Single-group time-series design Two samples Nonequivalent control group posttest only design Nonequivalent control group pre/posttest design Multiple-group time-series design

29 Chapter 13 Cross-sectional design Ss at different ages at one testing time Pro: fast and cheap Con: possible cohort effect Longitudinal design Ss repeatedly tested over time Pro: examine change (age effect) Con: attrition Sequential design Mix of cross-sectional and longitudinal Pro: benefits of both cross/longitudinal designs Con: statistics are difficult/unknown

30 Schaie, K. W. (1994). The course of adult intellectual development. American Psychologist, 49,

31 Figure. Cross-Sectional Mean T Scores for Single Markers of the Primary Mental Abilities

32 Figure 3. Longitudinal Estimates of Mean T Scores for Single Markers of the Primary Mental Abilities

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