Quiz 3:Ch 8- Chi Square Goodness of Fit Test

Calculating odds ratio for goodness of fit test: (effect size)

(probability of event occuring)/ (probability of event NOT occuring) pr yes given event/pr no given event ex) If I tell you that there is a 75% chance that we will have a pop quiz next week, what are the odds of us having one? p= 0.75 and 1 -p= 0.25 .75/.25=3 *there is a 3:1 odds ratio of us having a pop quiz*

expected frequencies, Alpha level, degrees of freedom, chi distribution, and critical value of goodness of fit test

*expected frequencies:* *expected frequencies* can be found by: n(expected proportion in fraction or decimal form given by the model) -where n is the sample size *Alpha:* 0.05 and is ALWAYS right tailed (because of the square in chi-squared test) *degrees of freedom:* # of categories-1 (aka # of rows in table -1) ---- df can tell you *critical number* from chi square table ---- df can tell you what *chi distribution graph* to use for the test statistic

what does a goodness of fit test do?

-allows us to handle categorical and discrete numerical variables having more than two outcomes. -It also allows us to assess the fit of more complex probability models. -allows us to test whether there's a statistically difference between a claimed model of the population to a sample of the population.

Steps for running a chi square goodness of fit test:

1.State the hypotheses 2.Calculate expected counts (under H0) and check assumptions 3. State alpha, degrees of freedom, note chi distribution being used given your df, and Calculate the test statistic and compare to the critical value. 4.Make a conclusion in context, citing the appropriate statistics (𝜒!, df, p-value/alpha)

goodness of fit test

A statistical test of the hypothesis that data have been randomly sampled or generated from a population that follows a particular *theoretical distribution or model*. The most common such tests are chi-square tests. SINGLE CATEGORICAL VARIABLE

null and alternative hypothesis for chi goodness of fit test:

H0: the distribution of ________ DOES NOT differ from the proposed model of _________ Ha: the distribution of ________ DOES differ from the proposed model of _________

decision rule for chi square goodness of fit test

REJECT NULL IF: If χ2> critical, then p-value < α FAIL TO REJECT NULL IF: If χ2< critical, then p-value > α

chi-square goodness of fit test

Statistical test used to evaluate how well a set of observed values fit the expected values. *The probability associated with a calculated chi-square value is the probability that the differences between the observed and the expected values may be due to chance.* determines whether *a single categorical variable* has a specified distribution expressed as the proportion of individuals falling into each possible category *compares frequency data to a probability model stated by the null hypothesis.*

what does the chi square goodness of fit test do?

Tests whether the distribution of counts in one categorical variable matches the distribution predicted by a model Inference for a single categorical variable TRIES TO ANSWER THIS QUESTION: Does the distribution significantly differ from some claimed (null) distribution? *Compares the observed counts (from a random sample) of a single categorical variable to what we would expect from the null distribution*

Chi-distribution

The theoretical distribution that models the test statistic for doing Chi-Square tests

test statistic for goodness of fit test

The χ 2 statistic measures the discrepancy between observed frequencies from the data and expected frequencies from the null hypothesis. χ 2 calculations use the absolute frequencies (i.e., counts) for the observed and expected frequencies, not proportions or relative frequencies. Using proportions in the calculation of χ 2 will give the wrong answer. USE WHOLE NUMBERS WITH DECIMALS (if decimals are necessary) Remember that if the data exactly matched the expectation of the null hypothesis, χ 2 would be zero.

proportional model

a simple probability model in which the frequency of occurrence of events is proportional to the number of opportunities ex) all babies born in the united states are born evenly throughout the week. (test this claim)

assumptions for a chi square goodness of fit test:

•Data is from a random sample •Cases must be individual and independent observations •Must have a sufficient sample size o All expected counts ≥ 1 o At least 80% of expected counts ≥ 5 *expected frequencies* can be found by: n(expected proportion in fraction or decimal form given by the model) -where n is the sample size ■ None of the categories should have an expected frequency less than one. ■ No more than 20% of the categories should have expected frequencies less than five. Notice that these restrictions refer to the expected frequencies, not to the observed frequencies. If these conditions are not met, then the test becomes unreliable. If one of these conditions is not met, then we have two options. One option, if possible, is to combine some of the categories having small expected frequencies to yield fewer categories having larger expected frequencies (remember to change the degrees of freedom accordingly). We can do this only if the new combined categories make biological sense. A second option is to find an alternative to the χ 2 goodness-of fit test

Quick summary of chapter from back of the textbook:

■ The χ 2 goodness-of-fit test compares the frequency distribution of a discrete or categorical variable with the frequencies expected from a probability model. ■ Goodness of fit is measured with the χ 2 test statistic. ■ The χ 2 test statistic has a null distribution that is approximated by the theoretical χ 2 distribution. The approximation is excellent as long as no expected frequencies are less than one and no more than 20% of the expected frequencies are less than five. It may be necessary to combine categories to meet these criteria. ■ The theoretical χ 2 distribution is a continuous distribution. Probability is measured by the area under the curve. ■ The null hypothesis is rejected at significance level α if the observed χ 2 statistic exceeds the critical value of the χ 2 distribution corresponding to α. ■ Under the proportional probability model, events fall in different categories in proportion to the number of opportunities. Rejecting H0 implies that the probabilities are not proportional.