When to use them? When the dependent variable is nominal Two different chi-square tests Chi square test for goodness-of-fit

Transcription

1 When to use them? When the dependent variable is nominal Two different chi-square tests Chi square test for goodness-of-fit One nominal variable Chi square test for independence Two nominal variables Chi square test for goodness-of-fit One nominal variable No IV or DV Has two or more categories It tests for whether there is a good fit with the null hypothesis or not If there is a bad fit it will support the research/experimental hypothesis Example Are there more babies born in the Spring rather than the Autumn in cold places? Observations of 19 babies born in the Autumn and 50 babies born in the Spring are given- this sample was taken from Alaska Conduct a test on these data The main point of a Are these numbers different than what we would expect by chance? If so, are they different enough to be called significantly different Writing your hypotheses Null: There is no difference in the timing of births for babies born in Alaska and the timing of births for babies born in the general population. Could also say: Alaska babies have the same timing for births of babies as the general population Research: There is a difference between the timing of births for babies born in Alaska and the timing of births for babies born in the general population

2 Step 3: Determining the characteristics Your book claims that the only thing to do here is df df = k-1 (k = number of categories) df = -1 = 1 Determining critical values Go to a table Critical value is Calculating the Category Observed Expected O-E (O-E) (O-E) /E Spring 50 Autumn 19 The most difficult part is calculating the expected values Given that there were 44 babies born in the sample ( ), what would we expect if there was no difference between our sample and the general population? 44 / = 1 Category Observed Expected O-E (O-E) (O-E) /E Spring 50 1 Autumn 19 1 Then complete the rest of the math here And then add the last column up Category Observed Expected O-E (O-E) (O-E) /E Spring Autumn So our Chi Square = 7.6 The numbers in the rows of the table are NOT always the same- ours just were in this case 7.6 > 3.841, so we will reject the null hypothesis here It appears that people in cold weather have different birth timing patterns than the general population

3 Chi-square: Another Example You may have heard, Stay with your first answer on a multiple-choice test. Is changing answers more likely to be helpful or harmful? To examine this, Best (1979) studied the responses of 61 students in an introductory psychology course. He recorded the number of right-towrong, wrong-to-right, and wrong-to-wrong answer changes for each student. More wrong-to-right changes than right-to-wrong changes were made by 195 of the students, who were thus helped by changing answers; 7 students made more right-to-wrong changes than wrongto-right changes and thus hurt themselves. Using a.05 level of significance, test the hypothesis that the proportions of right-to-wrong and wrong-to-right changes are equal. State the research hypothesis and the null hypothesis Calculate df Observed Frequency The obtained frequency for each category. right-to-wrong wrong-to-right Observed Expected Frequency The hypothesized frequency for each distribution, given the null hypothesis is true. Expected proportion multiplied by number of observations. right-to-wrong wrong-to-right Expected.5* = 111.5* = 111 Set the Critical value α =.05 df = 1 χ crit = 3.84 Calculate the test statistic. χ = right-to-wrong wrong-to-right Observed Expected ( f o f f e e ) (7 111) ( ) χ = = = = Decide if your result is significant. Reject H0, 17.14>3.84 Interpret your results. The proportion of right-to-wrong changes and wrong-to-right changes is not equal.

4 The other chi-square Chi-square for independence Two nominal variables involved Is one variable independent of the other? That is does one influence the other or are they completely independent events Latane and Bidwell (1977) Examined social affiliation Wanted to look at males and females Either in groups or not in groups Notice that there are two nominal variables here Gender Grouping So this is a case for the Chi-square test of independence How many different end results will we end up with? Draw a table for it. Writing the hypotheses Null: Grouping is independent of whether one is male or female Research/experimental: Grouping depends upon whether one is male or female Step 3: Characteristics Again, just the df here Step 4: Critical values Again, is our critical value df x = (df row )(df column ) df row = k row 1 df column = k column - 1 1x1 = 1

5 Step 5: Calculating the statistic Here is our data Take a look at it- does it appear that there is a difference between males and females in groups or alone? In groups Male 4 36 Female Alone With two variables, it is slightly more complicated First we calculate the totals for each of our observed values In groups Alone Total Male Female Total Then we compute the expected frequencies based on those observed values Observed In groups Alone Total Male Female Total Notice that they are not all the same This is because they didn t observe the same number of males and females in that sample Their research question wasn t are there more males than females, but instead was about whether males and females affiliate differently Expected In groups Alone Total Male 60*60/14 60*8/14 60 Female 60*8/14 8*8/14 8 Total Expected In groups Alone Total Male Female Total Final calculations Category O E O-E (O-E) (O-E) /E Male-alone Malegroups Femalegroups Femalealone Add them all up..1 = chi square.1<3.841, so we will fail to reject the null Overview Cannot be negative because all discrepancies are squared. Will be zero only in the unusual event that each observed frequency exactly equals the corresponding expected frequency. Other things being equal, the larger the discrepancy between the expected frequencies and their corresponding observed frequencies, the larger the observed value of chisquare. It is not the size of the discrepancy alone that accounts for a contribution to the value of chi-square, but the size of the discrepancy relative to the magnitude of the expected frequency. The value of chi-square depends on the number of discrepancies involved in its calculation.

6 Example 1 Do people that move farther away from home end up having a more exciting life? Exciting Routine Dull Same City Same state/diff city Diff State Example Observed Frequency The obtained frequency for each category. f o