DFTopic (13) Further tests for Univariate Data 13-1 Topic (13) Further Tests for Single Populations A) CATEGORICAL DATA – Chi-Square Tests For Univariate Data Recall that a categorical variable is one in which the possible values are categories or groupings. We’ve seen one such variable: it’s the binary variable with only two possible outcomes: success or failure. We also learned how to test hypothesis about the proportion of successes in a population of binary values. In this topic we explore testing hypotheses about categorical variables with MORE than two outcomes. EXAMPLE Consider an experiment in which two different tomato phenotypes are crossed and the resulting offspring observed. The parent types are tall cut-leaf tomatoes and dwarf potato-leaf tomatoes. Variable: Offspring Phenotype Possible Values: 1) tall cut-leaf, 2) tall potato-leaf, 3) dwarf cut-leaf, and 4) dwarf potato-leaf. If Mendel’s laws of inheritance hold, the resulting population proportions in the offspring would be 1) , 2) , 3) 16/9 16/3 16/3 , and 4) 16/1 . One might hypothesize that Mendel’s Laws don’t hold for theseTopic (13) Further tests for Univariate Data 13-2 genes. In an experiment to test that, the researcher observed the proportions 1) 0.575, 2) 0.179, 3) 0.182, and 4) 0.065 based on a sample of 1611 offspring. n=1611 Tall Cut Tall Potato Dwarf Cut Dwarf Potato Observed Proportion 0.5750 0.1790 0.1820 0.0650 Expected Proportion 0.5625 0.1875 0.1875 0.0625 EXAMPLE Consider an observational study in which the types of insects that feed on the nectar from a certain flower are studied. The scientist randomly selects hours during the day over several days during the summer season and selects several different plants. She counts the number of different kinds of insects that feed at the plant during the study. Variable: Insect Family Possible Values: 1) bees, 2) wasps, or 3) flies One might hypothesize that this flower attracts the different insect families in unequal proportions. Important Point: Testing procedures for hypotheses of this form are called Goodness-of-Fit tests.Topic (13) Further tests for Univariate Data 13-3 These tests compare the sample proportions to the hypothesized proportions to see how “good the fit is”. Important Point: These categories must be mutually exclusive and exhaustive. Notation: k = number of possible categories that the variable of interest can have. Category True Population Proportion Sample Proportion HypothesizedPopulation Proportion 1 1π 1p 01π 2 2π 2p 02π … … … … k kπ kp 0kπ “Exhaustive” means that ∑=1iπ, ∑=1ip, and ∑= 10iπ. “Exclusive” means that each sampling unit can be put into ONLY one category.Topic (13) Further tests for Univariate Data 13-4 EXAMPLE tomatoes and Mendel’s Laws . k = 4 Category True Population Proportion Sample Proportion Hypothesized Population Proportion tall cut-leaf 1π 575.01=p 16901=π tall potato 2π 179.02=p 16302=π dwarf cut 3π 182.03=p 16303=π dwarf potato 4π 065.04=p 16104=π Now, for a sample of size n and a set of hypothesized proportions under the null hypothesis, I can calculate how many sample units should be in each category (if there was no sampling variability, of course). These numbers are called the EXPECTED CELL COUNTS under the null hypothesis and are calculated as n×hypothesized value (0iπ) for that category (cell). The OBSERVED CELL COUNTS are the actual counts seen in each category during the experiment. Category Expected ObservedTopic (13) Further tests for Univariate Data 13-5 Count Count 1 01πn 1np 2 02πn 2np … … … k 0kπn knp Important Point: The following test procedure is valid only if the sample sizes and hypothesized proportions are such that virtually every cell has an expected count of 5 or more. If they aren’t you must use a different test procedure. EXAMPLE tomatoes and Mendel’s Laws. n = 1611 Category Expected Count Observed Count tall cut-leaf 2.906)16/9(161101==πn9261=np tall potato 1.302)16/3(161102==πn2882=npdwarf cut 1.302)16/3(161103==πn2932=npdwarf potato 7.100)16/1(161104==πn 104=knpTopic (13) Further tests for Univariate Data 13-6 Hypotheses: Ho: 16/91=π , 16/32=π , 16/33=π, and 16/14=π HA: not Ho (Ho is not true) Important Point: Note how uninformative the alternative hypothesis is in a goodness-of-fit test. These tests compare the sample data against a specific set of hypothesized proportions. If the null hypothesis is rejected, one cannot tell what the true proportions are, only that they are not the ones listed in the null hypothesis. Significance Level: let’s choose α=0.04. Test Statistic: is a summary of the comparison of the observed and expected cell counts. The actual form is ∑=cells all22count expectedcount) expected-count (observedχ This is called the CHI-SQUARE or GOODNESS-OF-FIT STATISTIC. Important Point: the closer the expected and observed counts are to each other, the smaller the value of . So, 2χTopic (13) Further tests for Univariate Data 13-7 small values of support the null hypothesis and large values support H2χA. EXAMPLE tomatoes and Mendel’s Laws. Category Expected Count Observed Count EEO2)( − tall cut-leaf 2.90601=πn9261=np 0.433 tall potato 1.30202=πn2882=np0.658 dwarf cut 1.30203=πn2932=np0.274 dwarf potato 7.10004=πn104=knp0.108 So, 473.1108.274.658.433.2=+++=χ P-value: under the null hypothesis, the test statistic has a sampling distribution known as the CHI-SQUARE DISTRIBUTION. Like the T-distribution, the shape of the Chi-Square Distribution depends on the degrees of freedom. Here, df = k – 1. 2χ Important Point: the degrees of freedom for the Chi-Square Goodness of Fit test are the number of categories minus 1 NOT the sample size minus 1.Topic (13) Further tests for Univariate Data 13-8 The p-value is the area under the Chi-square distribution to the right of the test statistic value: To find the P-value, first calculate and the df. Go to the table at the end of this topic. 2χ Find the row labeled with the df you have for your test. Go across the row, until you find two values that bracket your value. The P-value for your test-statistic lies between the two column headings above your bracketing values. 2χ2χ EXAMPLE
View Full Document
Unlocking...