CHAPT 18 Resampling and Nonparametric Approaches to Data & Categorical Data and Chi (Kahy) -Square

PARAMETRIC TESTS PARAMETRIC tests are more accurate than NONPARAMETRIC tests and they have 3 assumptions (research). 1. Random selection 2. Independent of observation 3. Sample is taken from a normal population with a normal distribution

NONPARAMETRIC STATISTICAL TESTS or distribution-free tests These tests which also used for hypotheses testing are rely on less restrictive assumptions about populations than parametric tests. 1.Wilcoxon’s rank-sum test: This test is often thought of as the distribution-free analogue of the t-test for two independent sample. 2. The man-whitney U Test: This test is very similar to Wilcoxon’s rank-sum test

NONPARAMETRIC STATISTICAL TESTS or distribution-free tests 3. Kruskal-Wallis one way anova: This test is direct generalization of the Wilcoxon’s rank-sum test to the case in which we have three or more independent groups. As such it is the distribution-free analogue of one way anova.

NONPARAMETRIC STATISTICAL TESTS or distribution-free tests 4. Friedman’s rank test for K correlated samples: This test was developed by the well-known economist Milton Friedman. It is the distribution-free analogue of the one-way repeated-measures Anova. It is closely related to a standard repeated-measure ANVA applied to ranks instead of raw scores (see data on page 680).

5. CHI (Kahy) SQURE test for goodness of fit and Pearson CHI SQURE (special case)

CHI SQURE CHI SQURE is similar to frequency distribution (f ). The independent observation is the basic assumption for a chi-square hypothesis test. It is used for comparative studies (use ranking).

CHI SQURE Ex1. Of the two leading brands of cola, which is preferred by most Americans? Coke Pepsi

CHI-SQURE test for goodness of fit Ex2. which psychotherapy technique is used by most psychologists? e.g. Cognitive, Behavioral, Humanistic, CBT, Psychodynamics, etc.

CHI-SQURE test for goodness of fit

Problem 1 (one variable, eye color) Distribution of Eye colors for a sample of n=40 individuals. FYI  From the chapter 2 Frequency Distribution (X )eye colors (f ) frequencies Blue 12 Brown 21 Green 3 Other 4 see next slide for Chi square

Problem 1 Distribution of Eye colors for a sample of n=40 individuals. Frequency distribution Fo Blue 12 Brown 21 Green 3 Others 4

Problem 1 Distribution of Eye colors for a sample of n=40 individuals. Frequency distribution Fo n=40 Fe Blue 12 Brown 21 Green 3 Others 4 Blue 10 Brown Green Others

H0 : fo = fe no preference for any specific eye color The question for the hypothesis test is whether there are any distribution of the four possible eye colors. Are any of the eye color distribution would be expected simply by chance? We will set alpha at α =.05 Step 1) H0 : fo = fe no preference for any specific eye color H1 : fo ≠ fe preference for specific eye color

Problem 1 Distribution of Eye colors for a sample of n=40 individuals. Frequency distribution fo fe Blue 12 Brown 21 Green 3 Others 4 10

Step 2 df=(R-1)(C-1)

Problem 2 A researcher examining art appreciation selected an abstract painting that had no obvious top or bottom. Hangers were placed on the painting so that it could be hung with any one of the four sides at the top. The painting was shown to a sample of n=50 participants, and each was asked to hung the painting in the orientation that looked correct. The following data indicate how many people choose each of the four sides to be placed at the top. fo Top up (correct) 18 Bottom up 17 Left side up 7 Right side up 8

Problem 2 fe fo 18 17 7 8 12.5 Top up (correct) Bottom up Left side up n= 50 fe Top up (correct) 18 Bottom up 17 Left side up 7 Right side up 8 Top up (correct) 12.5 Bottom up Left side up Right side up

Problem 2 The question for the hypothesis test is whether there are any preferences among the four possible orientations. Are any of the orientations selected more (or less) often than would be expected simply by chance? We will set alpha at α =.05

Problem 2 Step 1) H0 : fo=fe no preference for any specific orientation H1 : fo≠fe preference for specific orientation

Step 2 df=(R-1)(C-1)

The value of chi-square cannot be negative. if the number of categories is increased, the critical value will decrease.

c2= Σ(fo-fe)² /fe fo= observed frequency fe= expected frequency df=(R-1)(C-1) R is usually 2 for Chi-square test for independence (2 -1)=1 R= # of Rows and C= # of Columns df=C-1 fo = H1 fe = n/c = Ho or fe = pn where p is the proportion stated in the Null Hypothesis and n is the sample size.

Problem 3 Ten years ago, only 15% (convert to proportion) of the U.S. population consisted of people more than 65 years old. A researcher plans to use a sample of n = 300 people to determine whether the population distribution has changed during the past ten years. If a chi-square test is used to evaluate the data, what is the expected frequency for the older-than-65 category?

Problem 3 Ten years ago, only 15% of the U.S. population consisted of people more than 65 years old. A researcher plans to use a sample of n = 300 people to determine whether the population distribution has changed during the past ten years. If a chi-square test is used to evaluate the data, what is the expected frequency (fe) for the older-than-65 category? Answer: 0.15X300=45 fe=p(n)

Problem 4 A chi-square test for goodness of fit has df = 2. How many categories were used to classify the individuals in the sample?

Problem 4 A chi-square test for goodness of fit has df = 2. How many categories were used to classify the individuals in the sample? df=C-1 2=C-1 C=3

Problem 4.1 A researcher used a sample of n = 60 individuals to determine whether there are any preferences among four brands of pizza. Each individual tastes all four brands and selects his/her favorite. If the data are evaluated with a chi-square test for goodness of fit using a = .05, then how large does the chi-square statistic need to be to reject the null hypothesis?

Problem 4.1 A researcher used a sample of n = 60 individuals to determine whether there are any preferences among four brands of pizza. Each individual tastes all four brands and selects his/her favorite. If the data are evaluated with a chi-square test for goodness of fit using a = .05, then how large does the chi-square statistic need to be to reject the null hypothesis? Answer:>7.82 see the critical values for chi-square on p. 691

Problem 5 Very tricky question A sample of 100 people is classified by gender (male/female) and by whether or not they are registered voters. The sample consists of 80 females and 20 males, and has a total of 60 registered voters. If these data are used for a chi-square test for independence, what is the total number of females not registered for the expected frequencies?

Problem 5 fR=20 fR = 80 Registered Not registered fC = 60 fc=40 FEMALES fR = 80 MALES fR=20

Problem 5 Very tricky question A sample of 100 people is classified by gender (male/female) and by whether or not they are registered voters. The sample consists of 80 females and 20 males, and has a total of 60 registered voters. If these data are used for a chi-square test for independence, what is the total number of females not registered for the expected frequencies? Answer: p=80/100=0.80 60 registered voters means 40 not registered voters. fe=p(n) 0.80X40=32

THE CHI-SQUARE TEST FOR INDEPENDENCE (Pearson CHI SQURE) The chi square statistic may also be used to test whether there is a relationship between two variables. In this situation, each individual in the sample is measured or classified on two separate variables. If the number of their categories don’t match then use df=(C1-1)(C2-1) ex. next

Problem 6 A chi-square test for independence is being used to evaluate the relationship between two variables, one of which is classified into three categories and the second of which is classified into four categories. The chi-square statistic for this test would have df equal to?

Problem 6 A chi-square test for independence is being used to evaluate the relationship between two variables, one of which is classified into three categories and the second of which is classified into four categories. The chi-square statistic for this test would have df equal to? Answer df=6 df=(C1-1)(C2-1) = (3-1)(4-1)=2X3=6

Problem 7 A sample of 100 people is classified by gender (male/female) and by whether or not they are registered voters. The sample consists of 80 females and 20 males, and has a total of 60 registered voters. If these data were used for a chi-square test for independence, what is the expected frequency for males who are registered voters?

Problem 7 fR=20 fR = 80 Registered Not registered fC = 60 fc=40 FEMALES fR = 80 MALES fR=20

Problem 7 A sample of 100 people is classified by gender (male/female) and by whether or not they are registered voters. The sample consists of 80 females and 20 males, and has a total of 60 registered voters. If these data were used for a chi-square test for independence, what is the expected frequency for males who are registered voters? Answer: p=20/100=0.20 fe=p(n) = 0.20X60=12

Problem 7.5 A sample of 100 people is classified by gender (male/female) and by whether or not they are registered voters. The sample consists of 80 females and 20 males, and has a total of 60 registered voters. If these data were used for a chi-square test for independence, what is the expected frequency for males who are NOT registered voters? Answer: p=20/100=0.20 60 registered voters means 40 not registered voters. fe=p(n) 0.20X40=8

Problem 7.6 A researcher is using a chi-square test for independence to examine the relationship between TV preferences and gender for a sample of n = 100 children. Each child is asked to select his/her favorite from a fixed set of three TV shows and each child is classified as male or female. The chi-square statistic for this study would have df equal to …..

Problem 7.7  

Problem 7.8 A chi-square test for independence has df = 3. What is the total number of categories (cells in the matrix) that were used to classify individuals in the sample?

Problem 8 A chi-square test for independence has df = 3. What is the total number of categories that were used to classify individuals in the sample? Answer: 8 df= (R-1)(C - 1) 3= (2-1)(4 - 1)

Problem 9 A researcher is conducting a chi-square test for independence to evaluate the relationship between gender and preference for three different designs for a new automobile. Each individual in a sample of n = 30 males and n = 30 females selects a favorite design from the three choices. If the researcher obtains a chi-square statistic of c2 = 4.81, what is the appropriate statistical decision for the test? Try a = .05 and then a = .01

Hypotheses testing (step 1) Ho: fo=fe H1: fo ≠ fe SPSS: Asymp. Sig is an abbreviation for asymptotic significance, which means that the significance is very close to 0 because you're WAY out in the tail of the test! Hypotheses testing (step 1)

Step 2: Locate the Critical region df=(R-1)(C-1)

Step 3

Step 4

Effect size use Phi-Coefficient use for 2 by 2 data matrix

Effect size=Cramer‘s V use for more than 2X2 data matrix For Cramer’s V, the value of df* is the smaller of either (R-1) or (C-1).

Effect size=Cramer‘s V use for more than 2X2 data matrix Under what conditions is Cramér’s V used to measure effect size for a chi-square test for independence? Answer: When either of the two variables consists of more than two categories

Special case of chi- square statistic next

THE CHI-SQUARE TEST FOR INDEPENDENCE (Pearson CHI SQURE) The chi square statistic may also be used to test whether there is a relationship between two variables. In this situation, each individual in the sample is measured or classified on two separate variables. Example next: fe=(fc.fR)/n

Problem 10 A researcher selects a sample of 100 people to investigate the relationship between gender (male/female) and registering to vote (registered/not registered). The sample consists of 40 females, of whom 30 are registered voters, and 60 males, of whom 40 are registered voters. If these data were used for a chi-square test for independence, what is the expected frequency for registered females?

Problem 10 A researcher selects a sample of 100 people to investigate the relationship between gender (male/female) and registering to vote. The sample consists of 40 females, of whom 30 are registered voters, and 60 males, of whom 40 are registered voters. If these data were used for a chi-square test for independence, what is the expected frequency for registered females? Answer: 28 fe=(fc.fR)/n see next slide fe=(70x40)/100=28

Problem 10 40 20 fR=60 30 10 fR = 40 registered not registered fC = 70 fc=30 FEMALES 30 10 fR = 40 MALES 40 20 fR=60

Problem 11 Pearson CHI SQURE Males and females Willingness to use mental health services (Hypotheses testing) PROBABLY NO MAYBE PROBABLY YES 30 75 45 MALES 17 32 11 FEMALES 13 43 34

Problem 11 Males and females Willingness to use mental health services (Hypotheses testing) PROBABLY NO MAYBE PROBABLY YES 30 75 45 MALES 17 32 11 FEMALES 13 43 34

SPSS Click on Data  Weight Casesy Weight Cases by fo  Ok Click Analyze Descriptive Statistics Crosstabs R Row C Column Do not move the fo Click on Statistics Chi Square Continue  Cick OK

Problem 12 An insurance investigator has observed the people with some astrological signs tend to be safer drivers than the people with other signs. Using insurance records, the investigator classified 200 people according to their astrological signs and whether or not they were involved in a car accident during the previous 12 months. The data are as follows: 3X2 Accident n=40 No Accident N=160 Total=200 Do the data indicate a significant relationship between sign and accidents? Test at the .05 level of significance. 10 20 70 40 50