Non-parametric Statistics. What We Will Cover in This Section. What Is/Are It?

Transcription

1 Non-parametric Statistics /9/ P766 Non-parametric statistics What We Will Cover in This Section Introduction Correlational Techniques Spearman Point Biserial Phi Coefficient Chi Square /9/ P766 Non-parametric statistics What Is/Are It? Statistical techniques applied to a set of variables when one cannot assume that the data are normally distributed or the data do not meet the requirements for an interval scale. /9/ P766 Non-parametric statistics 3

2 Correlational Techniques Technique Variable Variable Spearman Correlation (r s ) Point Biserial (r pb ) Ordinal data Interval or Ratio Ordinal data Dichotomous Phi Coefficient (N) Dichotomous Dichotomous /9/ P766 Non-parametric statistics Spearman Correlation (r s ) The Null Hypothesis is H o : D s = The Alternative Hypothesis is H : D s /9/ P766 Non-parametric statistics Spearman Correlation Example A researcher wanted to know if there was a relationship between leadership skill and aggressiveness. The investigator first had ten supervisors rank ordered on their leadership skill, then had them rank ordered on their aggressiveness. /9/ P766 Non-parametric statistics 6

3 Spearman Correlation Example Subject R agg R lead /9/ P766 Non-parametric statistics 7 d d Formula and Computation 6( Σd ) rs = N(N ) 6(3) r s = ( ) 9 rs = 99 r s =.86 /9/ P766 Non-parametric statistics 8 When Used Both sets of data are ordinal. or One set is interval/ratio and the other is ordinal. or The data are interval or ratio but are not normally distributed. /9/ P766 Non-parametric statistics 9

4 Assumptions. The sample has been randomly selected from the population.. The relationship between the variables is linear. /9/ P766 Non-parametric statistics Kool Facts Regarding r s. Given no tied ranks r s and r give the same result.. Pearson r is more sensitive. 3. Pearson r is a more powerful statistic. /9/ P766 Non-parametric statistics Point-Biserial (r pb ) Used when One variable is ordinal or interval The second variable has only two values. The Null Hypothesis is H o : D pb = The Alternative Hypothesis is H : D pb /9/ P766 Non-parametric statistics

5 Examples of r pb You want to correlate gender to GPA. You want to correlated drivers training to number of traffic accidents. You want to correlate marital status to depression. /9/ P766 Non-parametric statistics 3 Example Z.Z. Bottoms wanted to know if there were differences in attitudes toward rap-country-soul music. Z. Z. got a group of eight volunteers and divided them into two groups: those under 3 years old and those over 3 years old. Bottoms then had them rate their feelings about the music. These data are summarized on the following page. /9/ P766 Non-parametric statistics Summary Attitude 3 Age Group Over 3 Under 3 Over 3 Over 3 Under 3 Over 3 Under 3 Under 3 Age Category /9/ P766 Non-parametric statistics

6 Age Group by Evaluation (r pb =-.77) Evaluatio Age Group /9/ P766 Non-parametric statistics 6 Kool Facts Regarding r pb When the dichotomous variable is coded and, r pb is equal to the Pearson r. Point-biserial and t-test are closely related. r pb looks at strength and uses all data. t looks at differences between means. /9/ P766 Non-parametric statistics 7 Phi-coefficient (N) Used when Both variables are dichotomous. The Null Hypothesis is H o : N = The Alternative Hypothesis is H : N /9/ P766 Non-parametric statistics 8

7 Phi Example Pass Test? C o r r e c t No () Yes() Yes () 7 No () 9 6 N = -.3 /9/ P766 Non-parametric statistics 9 Chi-Square (χ ) /9/ P766 Non-parametric statistics One Way Chi-Square (χ ) Goodness of Fit One group. Group is grouped categorically (nominal scale). Null hypothesis. There is no difference in the distribution of the scores across the group. Alternative hypothesis. There is a difference in the scores across the group. /9/ P766 Non-parametric statistics

8 Chi-square Example The noted statistician, Dr. Anne Nova, was interested in academic preferences. She got a sample of students and asked them to select the school activity they liked best from the following list. Statistics Psychological Testing Experimental Psychology Recess /9/ P766 Non-parametric statistics Hypotheses If there is no specific prediction. Null Hypothesis. H o : f stat = f experimental = f testing = f recess Alternative Hypothesis. H A : f stat f experimental f testing f recess Note: The researcher may specify the proportion of cases in H A. /9/ P766 Non-parametric statistics 3 Results Statistics Experimental Testing Recess f e f o f o - f e (f o - f e ) (f o - f e ) f e /9/ P766 Non-parametric statistics /

9 Chi-Square Evaluation ( fo fe) χ =Σ fe χ = Degrees of freedom = k- Critical value (page 699) χ = 7.8, p <. crit /9/ P766 Non-parametric statistics Two-Way Chi-Square Test of Independence Two variables. People are grouped categorically (nominal scale). Null hypothesis. There is no difference in the distribution of the scores across the variables. Alternative hypothesis. There is a difference in the distribution of the scores across the variables. /9/ P766 Non-parametric statistics 6 Chi-Square (r x k) Example The noted statistician, Dr. Polly Nomial, wanted to repeat Anne Nova s study but was interested if there was an effect on the basis of gender. So she asked a sample of students to select their preference for academic activity then broke the group into male and female respondents. /9/ P766 Non-parametric statistics 7

10 Nomial s Data Statistics Experimental Testing Recess Total Male Female 3 9 /9/ P766 Non-parametric statistics 8. Compute the expected values. Statistics Experimental Testing Recess Total Male f e =9. f e =. f e = 8.33 f e =. Female f e =.6 f e = 8.89 f e = 6.67 f e = RowTotal xcolumntotal fe = Grand Total /9/ P766 Non-parametric statistics 9. Compute (f e - f o ) Male Statistics Experiment al Testing Recess f e =9. (f e -f o ) =9.7 f e =. (f e -f o ) =. f e = 8.33 (f e -f o ) =.79 f e =. (f e -f o ) =.3 Total Female f e =.6 (f e -f o ) =9.7 f e = 8.89 (f e -f o ) =.3 f e = 6.67 (f e -f o ) =.79 f e = 8.89 (f e -f o ) = /9/ P766 Non-parametric statistics 3

11 Male Female Statistics ( f e f o) 3. Compute f Experiment al Testing Recess f e =9. (f e -f o ) =9.7. f e =. (f e -f o ) =..9 f e = 8.33 (f e -f o ) = f e =. (f e -f o ) =.3.36 f e =.6 (f e -f o ) = f e = 8.89 (f e -f o ) =.3.38 f e = 6.67 (f e -f o ) =.79.8 f e = 8.89 (f e -f o ) =.3.7 e Total 3 9 /9/ P766 Non-parametric statistics 3 ( fe fo). Compute χ = Σ f e χ = df = (r-)(c-) What is the critical value? What is your conclusion? /9/ P766 Non-parametric statistics 3 Kool Facts About Chi-Square P values range from a low of, no difference, to higher values as the difference between f o and f e become larger. Degrees of freedom are based on the number of cells, not number of people. For complex contingency tables P does not indicate which cells are highly different. Post hoc tests have to be done to identify the differences. P does not give an indication as to how strong a relationship is. /9/ P766 Non-parametric statistics 33

12 x Contingency Table /9/ P766 Non-parametric statistics 3 Types of Chi-square. One-way chi-square. Single variable with multiple categories. Preference for car models. Choice of computer models. Compares the distribution of scores against a standard. Equal distribution of scores. A priori distribution of scores. Referred to as Goodness of fit. Similar to the One-way ANOVA. /9/ P766 Non-parametric statistics 3 Types of Chi-square. Independent Samples Two or more variables. School dropouts by ethnic group. Gender of rider by order of finish in a horse race. Called: Test of Independence. Similar to factorial ANOVA. /9/ P766 Non-parametric statistics 36

13 Types of Chi-square 3. Correlational surrogate. Detect a relationship between nominal variables. Single sample is measured on two variables. Classified into the groups on each variable. Examples. Personality type by class quartile. Type of drug abused by family marital status. /9/ P766 Non-parametric statistics 37 P Assumptions. The observations are independent; each observed frequency is generated by a different subject.. The observed cell frequencies are greater than. /9/ P766 Non-parametric statistics 38 /9/ P766 Non-parametric statistics 39