T-T ESTS AND A NALYSIS OF V ARIANCE Jennifer Kensler

O NE S AMPLE T-T EST

Used to test whether the population mean is different from a specified value. Example: Is the mean amount of soda in a 20 oz. bottle different from 20 oz?

S TEP 1: F ORMULATE THE H YPOTHESES The population mean is not equal to a specified value. H 0 : μ = μ 0 H a : μ ≠ μ 0 The population mean is greater than a specified value. H 0 : μ = μ 0 H a : μ > μ 0 The population mean is less than a specified value. H 0 : μ = μ 0 H a : μ < μ 0

S TEP 2: C HECK THE A SSUMPTIONS The sample is random. The population from which the sample is drawn is either normal or the sample size is large.

S TEPS 3-5 Step 3: Calculate the test statistic: Where Step 4: Calculate the p-value based on the appropriate alternative hypothesis. Step 5: Write a conclusion.

I RIS E XAMPLE 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 H 0 : μ = 3.5 cm H a : μ ≠ 3.5 cm

JMP Steps 2-4: JMP Demonstration Analyze  Distribution Y, Columns: Sepal Width Test Mean Specify Hypothesized Mean: 3.5

JMP O UTPUT Step 5 Conclusion: The sepal width is not significantly different from 3.5 cm.

T WO S AMPLE T-T EST

Two sample t-tests are used to determine whether the mean of one group is equal to, larger than or smaller than the mean of another group. Example: Is the mean cholesterol of people taking drug A lower than the mean cholesterol of people taking drug B?

S TEP 1: F ORMULATE THE H YPOTHESES The population means of the two groups are not equal. H 0 : μ 1 = μ 2 H a : μ 1 ≠ μ 2 The population mean of group 1 is greater than the population mean of group 2. H 0 : μ 1 = μ 2 H a : μ 1 > μ 2 The population mean of group 1 is less than the population mean of group 2. H 0 : μ 1 = μ 2 H a : μ 1 < μ 2

S TEP 2: C HECK THE A SSUMPTIONS 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.

S TEPS 3-5 Step 3: Calculate the test statistic where Step 4: Calculate the appropriate p-value. Step 5: Write a Conclusion.

T WO S AMPLE E XAMPLE A researcher would like to know whether the mean sepal width of a setosa irises is different from the mean sepal width of versicolor irises. Step 1 Hypotheses: H 0 : μ setosa = μ versicolor H a : μ setosa ≠ μ versicolor

JMP Steps 2-4: JMP Demonstration: Analyze  Fit Y By X Y, Response: Sepal Width X, Factor: Species

JMP O UTPUT Step 5 Conclusion: There is strong evidence (p- value < ) that the mean sepal widths for the two varieties are different.

P AIRED T-T EST

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.

S TEP 1: F ORMULATE THE H YPOTHESES The population mean difference is not equal to zero. H 0 : μ difference = 0 H a : μ difference ≠ 0 The population mean difference is greater than zero. H 0 : μ difference = 0 H a : μ difference > 0 The population mean difference is less than a zero. H 0 : μ difference = 0 H a : μ difference < 0

S TEP 2: C HECK THE ASSUMPTIONS The sample is random. The data is matched pairs. The differences have a normal distribution or the sample size is large.

S TEPS 3-5 Where d bar is the mean of the differences and s d is the standard deviations of the differences. Step 4: Calculate the p-value. Step 5: Write a conclusion. Step 3: Calculate the test Statistic:

P AIRED T-T EST E XAMPLE 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 H 0 : μ After - Before = 0 H a : μ After - Before > 0

P AIRED T-T EST E XAMPLE Steps 2-4: JMP Analysis: Create a new column of After – Before Analyze  Distribution Y, Columns: After – Before Test Mean Specify Hypothesized Mean: 0

JMP O UTPUT Step 5 Conclusion: There is not evidence that the fitness program increases flexibility.

O NE -W AY A NALYSIS OF V ARIANCE

O NE -W AY ANOVA ANOVA is used to determine whether three or more populations have different distributions. A B C Medical Treatment

ANOVA S TRATEGY 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.

O NE -W AY ANOVA M ODEL In other words, for each group the observed value is the group mean plus some random variation.

O NE -W AY ANOVA H YPOTHESIS Step 1: We test whether there is a difference in the means.

S TEP 2: C HECK ANOVA A SSUMPTIONS 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.

S TEP 3: ANOVA F T EST Compare the variation within the samples to the variation between the samples. A B C A B C Medical Treatment

ANOVA T EST S TATISTIC Variation within groups small compared with variation between groups → Large F Variation within groups large compared with variation between groups → Small F

MSG The mean square for groups, MSG, measures the variability of the sample averages. SSG stands for sums of squares groups.

MSE Mean square error, MSE, measures the variability within the groups. SSE stands for sums of squares error.

S TEPS 4-5 Step 4: Calculate the p-value. Step 5: Write a conclusion.

ANOVA E XAMPLE 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 H 0 : μ Drug A = μ Drug B = μ Drug C H a : The μ i are not all equal.

S TEPS 2-4 JMP demonstration Analyze  Fit Y By X Y, Response: Pain X, Factor: Drug

E XAMPLE 1: JMP O UTPUT AND C ONCLUSION Step 5 Conclusion: There is strong evidence that the drugs are not all the same.

F OLLOW -U P T EST The p-value of the overall F test indicates that 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).

T UKEY T ESTS Tukey’s test simultaneously tests JMP demonstration Oneway Analysis of Pain By Drug  Compare Means  All Pairs, Tukey HSD for all pairs of factor levels. Tukey’s HSD controls the overall type I error.

JMP O UTPUT The JMP output shows that drugs A and C are significantly different.

A NALYSIS OF C OVARIANCE

A NALYSIS O F C OVARIANCE (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

ANCOVA E XAMPLE Consider the previous example where we tested whether the patients receiving different drugs reported different levels of pain. Perhaps age and gender may influence the efficacy of the drug. We can use age and gender as covariates. JMP demonstration Analyze  Fit Model Y: Pain Add: Drug Age Gender

JMP O UTPUT

C ONCLUSION The one sample t-test allows us to test whether the mean of a group is equal to a specified value. The two sample t-test and paired t-test allows us to determine if the means of two groups are different. ANOVA and ANCOVA methods allow us to determine whether the means of several groups are statistically different.

SAS AND SPSS For information about using SAS and SPSS to do ANOVA:

R EFERENCES 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.