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T-TESTS AND ANALYSIS OF VARIANCE Jennifer Kensler Laboratory for Interdisciplinary Statistical Analysis Virginia Tech’s source for expert statistical analysis since 1948 www.lisa.stat.vt.edu Collaboration: Personalized statistical advice Great advice right now: Meet with LISA before collecting your data Short Courses: Designed to help graduate students apply statistics in their research Walk-In Consulting: Monday—Friday* 12-2PM for questions <30 minutes * Mon—Thurs in summer * We help with research—not class projects or homework Laboratory for Interdisciplinary Statistical Analysis Virginia Tech’s source for expert statistical analysis since 1948 www.lisa.stat.vt.edu Collaboration: Personalized statistical advice Great advice right now: Meet with LISA before collecting your data Short Courses: Designed to help graduate students apply statistics in their research Walk-In Consulting: Monday—Friday* 12-2PM for questions <30 minutes * Mon—Thurs in summer * We help with research—not class projects or homework T-TESTS AND ANALYSIS OF VARIANCE ONE SAMPLE T-TEST 5 ONE SAMPLE T-TEST Used to test whether the population mean is different from a specified value. Example: Is the mean height of 12 year old girls greater than 60 inches? 6 STEP 1: FORMULATE THE HYPOTHESES The population mean is not equal to a specified value. H0: μ = μ0 Ha: μ ≠ μ0 The population mean is greater than a specified value. H0: μ = μ0 Ha: μ > μ0 The population mean is less than a specified value. H0: μ = μ0 Ha: μ < μ0 7 STEP 2: CHECK THE ASSUMPTIONS The sample is random. The population from which the sample is drawn is either normal or the sample size is large. 8 STEPS 3-5 Step 3: Calculate the test statistic: y 0 t s/ n n yi y 2 Where s i 1 n 1 Step 4: Calculate the p-value based on the appropriate alternative hypothesis. 9 Step 5: Write a conclusion. IRIS EXAMPLE A researcher would like to know whether the mean sepal width of a variety of irises is different from 3.5 cm. The researcher randomly measures the sepal width of 50 irises. Step 1: Hypotheses H0: μ = 3.5 cm Ha: μ ≠ 3.5 cm 10 JMP Steps 2-4: JMP Demonstration Analyze Distribution Y, Columns: Sepal Width Test Mean Specify Hypothesized Mean: 3.5 11 JMP OUTPUT Step 5 Conclusion: The mean sepal width is not significantly different from 3.5 cm. 12 TWO SAMPLE T-TEST 13 TWO SAMPLE T-TEST Two sample t-tests are used to determine whether the population mean of one group is equal to, larger than or smaller than the population mean of another group. Example: Is the mean cholesterol of people taking drug A lower than the mean cholesterol of people taking drug B? 14 STEP 1: FORMULATE THE HYPOTHESES The population means of the two groups are not equal. H0: μ1 = μ2 Ha: μ1 ≠ μ2 The population mean of group 1 is greater than the population mean of group 2. H0: μ1 = μ2 Ha: μ1 > μ2 The population mean of group 1 is less than the population mean of group 2. H0: μ1 = μ2 15 Ha: μ1 < μ2 STEP 2: CHECK THE ASSUMPTIONS The two samples are random and independent. The populations from which the samples are drawn are either normal or the sample sizes are large. The populations have the same standard deviation. 16 STEPS 3-5 Step 3: Calculate the test statistic y1 y2 t 1 1 sp n1 n2 (n1 1) s12 (n2 1) s2 2 where sp n1 n2 2 Step 4: Calculate the appropriate p-value. Step 5: Write a Conclusion. 17 TWO SAMPLE EXAMPLE A researcher would like to know whether the mean sepal width of setosa irises is different from the mean sepal width of versicolor irises. Step 1 Hypotheses: H0: μsetosa = μversicolor Ha: μsetosa ≠ μversicolor 18 JMP Steps 2-4: JMP Demonstration: Analyze Fit Y By X Y, Response: Sepal Width X, Factor: Species 19 JMP OUTPUT Step 5 Conclusion: There is strong evidence (p- value < 0.0001) that the mean sepal widths for the two varieties are different. 20 PAIRED T-TEST 21 PAIRED T-TEST The paired t-test is used to compare the means of two dependent samples. Example: A researcher would like to determine if background noise causes people to take longer to complete math problems. The researcher gives 20 subjects two math tests one with complete silence and one with background noise and records the time each subject takes to complete each test. 22 STEP 1: FORMULATE THE HYPOTHESES The population mean difference is not equal to zero. H0: μdifference = 0 Ha: μdifference ≠ 0 The population mean difference is greater than zero. H0: μdifference = 0 Ha: μdifference > 0 The population mean difference is less than a zero. H0: μdifference = 0 Ha: μdifference < 0 23 STEP 2: CHECK THE ASSUMPTIONS The sample is random. The data is matched pairs. The differences have a normal distribution or the sample size is large. 24 STEPS 3-5 Step 3: Calculate the test Statistic: d 0 t sd / n Where d bar is the mean of the differences and sd is the standard deviations of the differences. Step 4: Calculate the p-value. Step 5: Write a conclusion. 25 PAIRED T-TEST EXAMPLE A researcher would like to determine whether a fitness program increases flexibility. The researcher measures the flexibility (in inches) of 12 randomly selected participants before and after the fitness program. Step 1: Formulate a Hypothesis H0: μAfter - Before = 0 Ha: μ After - Before > 0 26 PAIRED T-TEST EXAMPLE Steps 2-4: JMP Analysis: Create a new column of After – Before Analyze Distribution Y, Columns: After – Before Test Mean Specify Hypothesized Mean: 0 27 JMP OUTPUT Step 5 Conclusion: There is not evidence that the fitness program increases flexibility. 28 ONE-WAY ANALYSIS OF VARIANCE 29 ONE-WAY ANOVA ANOVA is used to determine whether three or more populations have different distributions. A B C 30 Medical Treatment ANOVA STRATEGY The first step is to use the ANOVA F test to determine if there are any significant differences among means. If the ANOVA F test shows that the means are not all the same, then follow up tests can be performed to see which pairs of means differ. 31 ONE-WAY ANOVA MODEL yij i ij Where yij is the responseof the jth trial on the ith factor level i is the mean of the ith group ij ~ N (0, 2 ) i 1,, r j 1, , ni In other words, for each group the observed value is the group mean plus some random variation. 32 ONE-WAY ANOVA HYPOTHESIS Step 1: We test whether there is a difference in the means. H 0 : 1 2 r H a : The i are not all equal. 33 STEP 2: CHECK ANOVA ASSUMPTIONS The samples are random and independent of each other. The populations are normally distributed. The populations all have the same variance. The ANOVA F test is robust to the assumptions of normality and equal variances. 34 STEP 3: ANOVA F TEST A B C A B C Medical Treatment Compare the variation within the samples to the 35 variation between the samples. ANOVA TEST STATISTIC Variation between Groups MSG F Variation within Groups MSE Variation within groups Variation within groups small compared with large compared with variation between groups variation between groups 36 → Large F → Small F MSG The mean square for groups, MSG, measures the variability of the sample averages. SSG stands for sums of squares groups. SSG MSG r -1 n1 ( y1 y ) 2 n 2 ( y2 y ) 2 n r ( y1 y ) 2 r -1 37 MSE Mean square error, MSE, measures the variability within the groups. SSE stands for sums of squares error. SSE MSE n-r (n1 - 1)s1 (n 2 - 1)s2 (n r - 1)s2 2 2 r n-r Where ni (y j 1 ij yi ) si ni 1 38 STEPS 4-5 Step 4: Calculate the p-value. Step 5: Write a conclusion. 39 ANOVA EXAMPLE A researcher would like to determine if three drugs provide the same relief from pain. 60 patients are randomly assigned to a treatment (20 people in each treatment). Step 1: Formulate the Hypotheses H0: μDrug A = μDrug B = μDrug C Ha : The μi are not all equal. 40 STEPS 2-4 JMP demonstration Analyze Fit Y By X Y, Response: Pain X, Factor: Drug 41 JMP OUTPUT AND CONCLUSION Step 5 Conclusion: There is strong evidence that the drugs are not all the same. 42 FOLLOW-UP TEST The p-value of the overall F test indicates that the level of pain is not the same for patients taking drugs A, B and C. We would like to know which pairs of treatments are different. One method is to use Tukey’s HSD (honestly significant differences). 43 TUKEY TESTS Tukey’s test simultaneously tests H 0 : i i ' H a : i i ' for all pairs of factor levels. Tukey’s HSD controls the overall type I error. JMP demonstration Oneway Analysis of Pain By Drug Compare Means All Pairs, Tukey HSD 44 JMP OUTPUT The JMP output shows that drugs A and C are significantly different. 45 TWO-WAY ANALYSIS OF VARIANCE 46 TWO-WAY ANOVA We are interested in the effect of two categorical factors on the response. We are interested in whether either of the two factors have an effect on the response and whether there is an interaction effect. An interaction effect means that the effect on the response of one factor depends on the level of the other factor. 47 INTERACTION No Interaction Interaction Factor B Low Factor B Low Factor B High Factor B High Response Low High Response Low High Factor A Factor A 48 TWO-WAY ANOVA MODEL yijk i j ( ) ij ijk Where yijk is the responseof the kth trial on the ith factor A level and the jth factor B level is the overall mean i is the main effect of the ith level of factor A j is the main effect of the jth level of factor B ( ) ij is the interaction effect of the ith level of factor A and the jth level of factor B ijk ~ N (0, 2 ) i 1, , a j 1,, b k 1,...,nij 49 TWO-WAY ANOVA EXAMPLE We would like to determine the effect of two alloys (low, high) and three cooling temperatures (low, medium, high) on the strength of a wire. JMP demonstration Analyze Fit Model Y: Strength Highlight Alloy and Temp and click Macros Factorial to Degree 50 JMP OUTPUT Conclusion: There is strong evidence of an interaction between alloy and temperature. 51 ANALYSIS OF COVARIANCE 52 ANALYSIS OF COVARIANCE (ANCOVA) Covariates are variables that may affect the response but cannot be controlled. Covariates are not of primary interest to the researcher. We will look at an example with two covariates, the model is yij i covariates ij 53 ANCOVA EXAMPLE Consider the one-way ANOVA example where we tested whether the patients receiving different drugs reported different levels of pain. Perhaps age and gender may influence the pain. We can use age and gender as covariates. JMP demonstration Analyze Fit Model Y: Pain Add: Drug Age 54 Gender JMP OUTPUT 55 CONCLUSION The one sample t-test allows us to test whether the population mean of a group is equal to a specified value. The two-sample t-test and paired t-test allow us to determine if the population means of two groups are different. ANOVA and ANCOVA methods allow us to determine whether the population means of several groups are statistically different. 56 SAS AND SPSS For information about using SAS and SPSS to do ANOVA: http://www.ats.ucla.edu/stat/sas/topics/anova.htm http://www.ats.ucla.edu/stat/spss/topics/anova.htm 57 REFERENCES Fisher’s Irises Data (used in one sample and two sample t-test examples). Flexibility data (paired t-test example): Michael Sullivan III. Statistics Informed Decisions Using Data. Upper Saddle River, New Jersey: Pearson Education, 2004: 602. 58

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