3003PSY Week 10 Mini Lecture 1 One Way Chi-square PDF

3003PSY Survey Design and Analysis in Psychology ONE-WAY CHI-SQUARE  CHI SQUARE (c2) uChi square is a test used on frequency data uChi square allows us to test research questions with categorial data  MEASUREMENT LEVELS nominal ordinal interval ratio...

3003PSY Survey Design and Analysis in Psychology ONE-WAY CHI-SQUARE  CHI SQUARE (c2) uChi square is a test used on frequency data uChi square allows us to test research questions with categorial data  MEASUREMENT LEVELS nominal ordinal interval ratio categorical continuous measures are measures the equal interval scale just names rank ordering differences with a true for things of something between zero point e.g. gender e.g. 1st/2nd scores can e.g. length place be treated as equal units e.g. degrees C  CHI SQUARE (c2) uChi square is a test used on frequency data uChi square allows us to test research questions with categorial data What do we mean by frequency data? A research example  CHI SQUARE (c2) uChi square is a test used on frequency data uChi square allows us to test research questions with categorial data uIn the one-way chi square we can ask research questions like u Are people/things distributed evenly across the categories of the variable? uWe can also test out other distributions (we will come back to this)  uchi-square analyses involve working out the expected likelihood of something happening (what frequencies you should get if the null hypothesis is true) and comparing it with the observed frequencies (what CHI-SQUARE frequencies you do get) TEST uchi-square tells you how well the expected and observed frequencies match … for this reason it is often referred to as a test for “goodness of fit” (do the observed frequencies “fit” the expected frequencies?)  2 χ =∑2 ( fo − fe) CALCULATION fe OF c2 fo = observed frequency in the data fe = expected frequency under the null S means sum over all the categories basic idea: c2 will be small if differences between fo and fe are small  ONEWAY CHI-SQUARE: EXAMPLE uWe have gathered a sample of 284 participants in a study Choc & banana 131 uWe asked these people on their chupa chup flavour preference Cola 92 uWhat questions might we ask of these data? Strawberry 33 Most basic: are some flavours more popular than others? In other words - are Vanilla 28 people distributed evenly across the levels of chupa chup flavour preference?  ONEWAY CHI-SQUARE: EXAMPLE uWhat if 284 participants were distributed evenly across the levels of the chupa chup flavour preference variable? Choc & banana 71 uThey would be distributed as at right Cola 71 uTHIS IS THE NULL: H0 Strawberry 71 Most basic: are some flavours more popular than others? In other words - are Vanilla 71 people distributed evenly across the levels of chupa chup flavour preference?  EXPECTED FREQUENCIES uwhat do we expect? what is the null? uuniformly distributed -> equiprobable distribution udistributed in accord with theory udistributed in accord with previous observed frequencies unormally distributed  ONEWAY CHI-SQUARE: EXAMPLE fo fe Choc & banana 131 71 Cola 92 71 Strawberry 33 71 Vanilla 28 71  ONEWAY CHI-SQUARE: EXAMPLE 2 fo fe χ =∑ 2 ( fo − fe) Choc & banana 131 71 = ( 131 – 71)2 /71 + fe (92 – 71)2 /71 + Cola 92 71 (33 – 71)2 /71 + (28 – 71)2 /71 Strawberry 33 71 = 50.70 + 6.21 + 20.33 + 26.04 Vanilla 28 71 = 103.28  ONEWAY CHI-SQUARE: EXAMPLE uSo, now what does this mean?! fo fe uIs 103.28 a cause for celebration or Choc & banana 131 71 mourning? Cola 92 71 Strawberry 33 71 Vanilla 28 71 COMPARING OUR OBTAINED STATISTIC AGAINST CHI- SQUARE DISTRIBUTION (NULL) H0: in the population, the distribution across categories = expected frequencies Therefore, any discrepancy = chance How big can the discrepancy be before we reject possibility of H0 being true? i.e., a discrepancy large enough that less Use the standard cutoff of.05 than 5% of samples exhibit it when the null hypothesis is true. SAMPLING DISTRIBUTION OF c2 c2 is a family of distributions dependent on df however, df depends on number of cells, not number of participants df = k – 1 c2 values always positive can increase c2obt by increasing sample size e.g., could double c2 value by doubling sample size but c2crit remains the same SAMPLING DISTRIBUTION OF c2 From Howell, D. C. (2007). Statistical Methods for Psychology (6th ed.). Boston: PWS -Kent Publishing Co. SO, IN PRACTICE… We only need the critical value of chi square for the degrees of freedom in our study critical value = value that marks.05 region under the null Calculating df for a one-way chi square df = k – 1, k = number of categories Four categories of chupa chup flavour k = 4, so the df = 3 SELECTED CHI-SQUARE CRITICAL VALUES degrees of freedom critical value for p =.05, H0: true 1 3.84 2 5.99 3 7.81 4 9.49 5 11.07 6 12.59 taken from Thompson, 7 14.07 C.M. (1941). Table of percentage points of the χ2 8 15.51 distribution. Biometrika, 32, 9 16.92 187-191. 10 18.31  OUR CONCLUSION uOur obtained value of 103.28 is certainly larger than the critical value of 7.81 (for df = 3) so we conclude that the sample is not consistent with a population in which units are distributed evenly across the categories of flavour preference This is a significant difference  OUR CONCLUSION uOur obtained value of 103.28 is certainly larger than the critical value of 7.81 (for df = 3) so we conclude that the sample is not consistent with a population in which units are distributed evenly across the categories of flavour preference This is a significant difference  EXPECTED FREQUENCIES ufor some variables, simple equivalency distributions under chance might be a useful benchmark against which to test ue.g., marketing and popularity of choice uin other cases, we might have a more plausible expected distribution against which to test our sample uthat is, we already know that the categories won't be exactly equal  EXPECTED FREQUENCIES uwhat do we expect? what is the null? uuniformly distributed -> equiprobable distribution udistributed in accord with theory udistributed in accord with previous observed frequencies unormally distributed  SUMMARY uChi square tests research questions for nominal (aka categorical) data uChi square tests use frequency data uThe one-way chi square test is also referred to as the Goodness of Fit chi-square uThe chi square compares obtained frequencies to expected frequencies as stated under the null hypothesis uTo test the significance of a chi square, the obtained value is compared to the critical values in the chi square distribution uThe degrees of freedom in chi square depend on the number of categories – not the sample size uThe one-way chi square can specify different null hypothesis uA common null hypothesis is the equiprobable distribution