Classical Hypothesis Testing in R. R can do all of the common analyses that are available in SPSS, including:
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1 Classical Hypothesis Testing in R R can do all of the common analyses that are available in SPSS, including:
2 Classical Hypothesis Testing in R R can do all of the common analyses that are available in SPSS, including: χ 2 tests
3 Classical Hypothesis Testing in R R can do all of the common analyses that are available in SPSS, including: χ 2 tests t-tests
4 Classical Hypothesis Testing in R R can do all of the common analyses that are available in SPSS, including: χ 2 tests t-tests ANOVA
5 Classical Hypothesis Testing in R R can do all of the common analyses that are available in SPSS, including: χ 2 tests t-tests ANOVA Regression
6 Outline Inference for a Proportion Chi-Squared Tests of Independence One-Sample t-test Two-Sample t-test Test for Homogeneity of Variance Analysis of Variance Tukey s HSD Tests Multifactor Analysis of Variance Multiple Regression Testing Nested Models
7 Inference for a Proportion Let s read in gssdata.csv. If you already have it read in, we don t have to do this again.
8 Inference for a Proportion > gss = read.csv(file.choose())
9 Inference for a Proportion The variable VOTE92 indicates whether the respondent voted in the 1992 presidential election.
10 Inference for a Proportion The variable VOTE92 indicates whether the respondent voted in the 1992 presidential election. Let s examine a contingency table of this variable.
11 Inference for a Proportion > foo <- table(gss$vote92) > foo DID NOT VOTE DK VOTED
12 Inference for a Proportion In this example, there were 999 respondents who voted out of 1500 total respondents.
13 Inference for a Proportion In this example, there were 999 respondents who voted out of 1500 total respondents. We can carry out a hypothesis test using the prop.test() function.
14 Inference for a Proportion > prop.test(999, 1500) 1-sample proportions test with continuity correction data: 999 out of 1500, null probability 0.5 X-squared = , df = 1, p-value < 2.2e-16 alternative hypothesis: true p is not equal to percent confidence interval: sample estimates: p 0.666
15 Inference for a Proportion Note that by default, The null hypothesis π = 0.5 is tested against the alternative hypothesis π 0.5.
16 Inference for a Proportion Note that by default, The null hypothesis π = 0.5 is tested against the alternative hypothesis π 0.5. A 95% confidence interval for π is computed.
17 Inference for a Proportion Note that by default, The null hypothesis π = 0.5 is tested against the alternative hypothesis π 0.5. A 95% confidence interval for π is computed. Both the test and CI employ a correction for continuity.
18 Inference for a Proportion Note that by default, The null hypothesis π = 0.5 is tested against the alternative hypothesis π 0.5. A 95% confidence interval for π is computed. Both the test and CI employ a correction for continuity. Like any other function in R, these defaults can be changed.
19 Inference for a Proportion For example, to test the null hypothesis that π = 0.6 against the alternative hypothesis π > 0.6 and a 99% CI (one-sided) for π, all without the correction for continuity.
20 Inference for a Proportion > prop.test(999, 1500, p = 0.6, alternative = "greater", conf.level = 0.99, correct = FALSE) 1-sample proportions test without continuity correction data: 999 out of 1500, null probability 0.6 X-squared = , df = 1, p-value = 9.055e-08 alternative hypothesis: true p is greater than percent confidence interval: sample estimates: p 0.666
21 Outline Inference for a Proportion Chi-Squared Tests of Independence One-Sample t-test Two-Sample t-test Test for Homogeneity of Variance Analysis of Variance Tukey s HSD Tests Multifactor Analysis of Variance Multiple Regression Testing Nested Models
22 Chi-Squared Tests of Independence The chisq.test() function will compute Pearson s chi-squared test statistics and the corresponding p-value.
23 Chi-Squared Tests of Independence The chisq.test() function will compute Pearson s chi-squared test statistics and the corresponding p-value. We will apply it to examine whether respondents from different age groups differ in the frequency with which they report their income level.
24 Chi-Squared Tests of Independence > my.test <- chisq.test(gss$age, gss$income) > my.test Pearson's Chi-squared test data: gss$age and gss$income X-squared = , df = 8, p-value < 2.2e-16
25 Chi-Squared Tests of Independence What happens when we issue the following commands? mytest$observed
26 Chi-Squared Tests of Independence What happens when we issue the following commands? mytest$observed mytest$expected
27 Chi-Squared Tests of Independence What happens when we issue the following commands? mytest$observed mytest$expected mytest$residuals
28 Chi-Squared Tests of Independence In the case that we may be worried about the chi-squared approximation to the sampling distribution of the statistic, we can instead, use simulation to compute an approximate p-value.
29 Chi-Squared Tests of Independence > chisq.test(gss$age, gss$income, simulate.p.value = TRUE, B = 1000) Pearson's Chi-squared test with simulated p-value (based on 1000 replicates) data: gss$age and gss$income X-squared = , df = NA, p-value =
30 Outline Inference for a Proportion Chi-Squared Tests of Independence One-Sample t-test Two-Sample t-test Test for Homogeneity of Variance Analysis of Variance Tukey s HSD Tests Multifactor Analysis of Variance Multiple Regression Testing Nested Models
31 One-Sample t-test The function to perform a one-sample t-test is t.test().
32 One-Sample t-test The function to perform a one-sample t-test is t.test(). Using the wages data, suppose we wanted to perform a one-sample t-test to examine whether the mean salary for all employees was different from $6000.
33 One-Sample t-test > library(foreign) > wages <- read.spss("wages.sav", to.data.frame = T) > t.test(wages$salary, mu = 6000, alternative = "two.sided", conf.level = 0.95) One Sample t-test data: wages$salary t = , df = 92, p-value = 6.45e-12 alternative hypothesis: true mean is not equal to percent confidence interval: sample estimates: mean of x
34 Outline Inference for a Proportion Chi-Squared Tests of Independence One-Sample t-test Two-Sample t-test Test for Homogeneity of Variance Analysis of Variance Tukey s HSD Tests Multifactor Analysis of Variance Multiple Regression Testing Nested Models
35 Two-Sample t-test The function to perform a two-sample t-test is also t.test().
36 Two-Sample t-test The function to perform a two-sample t-test is also t.test(). Suppose we wanted to perform a two-sample t-test to examine whether the mean salary for all male employees was different than the mean salary for all female employees.
37 Two-Sample t-test > t.test(salary ~ sex, data = wages, var.equal = TRUE) Two Sample t-test data: salary by sex t = , df = 91, p-value = 1.076e-08 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: sample estimates: mean in group FEMALE mean in group MALE
38 Two-Sample t-test R can also perform a paired test by setting the argument paired = TRUE
39 Two-Sample t-test R can also perform a paired test by setting the argument paired = TRUE The data can also be formatted so that the data for males is in one column and the data for females is in another.
40 Outline Inference for a Proportion Chi-Squared Tests of Independence One-Sample t-test Two-Sample t-test Test for Homogeneity of Variance Analysis of Variance Tukey s HSD Tests Multifactor Analysis of Variance Multiple Regression Testing Nested Models
41 Test for Homogeneity of Variance We are not given a homogeneity of variance test unless we specifically ask for it.
42 Test for Homogeneity of Variance We are not given a homogeneity of variance test unless we specifically ask for it. The levene.test() function is available in the car library.
43 Test for Homogeneity of Variance > library(car) > levene.test(salary ~ sex, data = wages) Levene's Test for Homogeneity of Variance Df F value Pr(>F) group
44 Test for Homogeneity of Variance The results for this test will differ from those given in SPSS.
45 Test for Homogeneity of Variance The results for this test will differ from those given in SPSS. R uses the modified Levene s test (which uses deviations from the median rather than from the mean).
46 Outline Inference for a Proportion Chi-Squared Tests of Independence One-Sample t-test Two-Sample t-test Test for Homogeneity of Variance Analysis of Variance Tukey s HSD Tests Multifactor Analysis of Variance Multiple Regression Testing Nested Models
47 Analysis of Variance To perform an analysis of variance in R, we will use the aov() function.
48 Analysis of Variance To perform an analysis of variance in R, we will use the aov() function. It is typical to assign this function to an object.
49 Analysis of Variance > my.model <- aov(salary ~ sex, data = wages)
50 Analysis of Variance We can then obtain the results of our ANOVA by applying the anova() function to our fitted model.
51 Analysis of Variance > anova(my.model) Analysis of Variance Table Response: salary Df Sum Sq Mean Sq F value Pr(>F) sex e-08 Residuals sex *** Residuals --- Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
52 Analysis of Variance What does the command plot(my.model) do?
53 Analysis of Variance If the design is unbalanced the default results from anova() will not match those from SPSS.
54 Analysis of Variance If the design is unbalanced the default results from anova() will not match those from SPSS. The default sum of squares in R is type I.
55 Analysis of Variance If the design is unbalanced the default results from anova() will not match those from SPSS. The default sum of squares in R is type I. The Anova() function from the car library can be used to obtain type II or II SS.
56 Outline Inference for a Proportion Chi-Squared Tests of Independence One-Sample t-test Two-Sample t-test Test for Homogeneity of Variance Analysis of Variance Tukey s HSD Tests Multifactor Analysis of Variance Multiple Regression Testing Nested Models
57 Tukey s HSD Tests If we had more than two groups and found significant differences, we could perform Tukey HSD post hoc tests.
58 Tukey s HSD Tests If we had more than two groups and found significant differences, we could perform Tukey HSD post hoc tests. The multcomp library contains many other multiple comparison tests.
59 Tukey s HSD Tests > TukeyHSD(my.model) Tukey multiple comparisons of means 95% family-wise confidence level Fit: aov(formula = salary ~ sex, data = wages) $sex diff lwr MALE -FEMALE upr p adj MALE -FEMALE
60 Tukey s HSD Tests We can also plot the simultaneous confidence intervals.
61 Tukey s HSD Tests We can also plot the simultaneous confidence intervals. We apply the plot() function to our Tukey results.
62 Tukey s HSD Tests > plot(tukeyhsd(my.model)) 95% family wise confidence level MALE FEMALE
63 Outline Inference for a Proportion Chi-Squared Tests of Independence One-Sample t-test Two-Sample t-test Test for Homogeneity of Variance Analysis of Variance Tukey s HSD Tests Multifactor Analysis of Variance Multiple Regression Testing Nested Models
64 Multifactor Analysis of Variance We can add additional factors into the model using the + operator in our formula.
65 Multifactor Analysis of Variance We can add additional factors into the model using the + operator in our formula. We can add interaction terms into the model using the : operator in our formula.
66 Multifactor Analysis of Variance We can add additional factors into the model using the + operator in our formula. We can add interaction terms into the model using the : operator in our formula. my.model <- aov(salary data=wages) sex + race + sex:race,
67 Outline Inference for a Proportion Chi-Squared Tests of Independence One-Sample t-test Two-Sample t-test Test for Homogeneity of Variance Analysis of Variance Tukey s HSD Tests Multifactor Analysis of Variance Multiple Regression Testing Nested Models
68 Multiple Regression The function we used for the simple linear regression model can be expanded to include multiple independent variables.
69 Multiple Regression The function we used for the simple linear regression model can be expanded to include multiple independent variables. We can use the same operators in the formula as multifactor ANOVA in the lm() function.
70 Multiple Regression The function we used for the simple linear regression model can be expanded to include multiple independent variables. We can use the same operators in the formula as multifactor ANOVA in the lm() function. Suppose we wanted to model salary using sex, experience, and the sex by experience interaction terms.
71 Multiple Regression > int.model <- lm(salary ~ sex + exper + sex:exper, data = wages)
72 Multiple Regression > summary(int.model) Call: lm(formula = salary ~ sex + exper + sex:exper, data = wages) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error (Intercept) sexmale exper sexmale :exper t value Pr(> t ) (Intercept) < 2e-16 *** sexmale e-07 *** exper * sexmale :exper Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
73 Multiple Regression > anova(int.model) Analysis of Variance Table Response: salary Df Sum Sq Mean Sq F value Pr(>F) sex e-09 exper sex:exper Residuals sex *** exper. sex:exper. Residuals --- Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
74 Outline Inference for a Proportion Chi-Squared Tests of Independence One-Sample t-test Two-Sample t-test Test for Homogeneity of Variance Analysis of Variance Tukey s HSD Tests Multifactor Analysis of Variance Multiple Regression Testing Nested Models
75 Testing Nested Models Suppose we wanted to fit a saturated model using the sex, education and experience variables (including all interactions) to model salary.
76 Testing Nested Models Suppose we wanted to fit a saturated model using the sex, education and experience variables (including all interactions) to model salary. What is the R command to do so?
77 Testing Nested Models > full.model <- lm(salary ~ sex + exper + educ + sex:exper + sex:educ + educ:exper + sex:educ:exper, data = wages)
78 Testing Nested Models How would we fit a main effects only model?
79 Testing Nested Models > me.model <- lm(salary ~ sex + exper + educ, data = wages)
80 Testing Nested Models We can test whether the additional terms in the full model are necessary by using anova() function.
81 Testing Nested Models We can test whether the additional terms in the full model are necessary by using anova() function. We simply provide this function with two models one of which is nested in the other.
82 Testing Nested Models > anova(me.model, full.model) Analysis of Variance Table Model 2: salary ~ sex + exper + educ + sex:exper + sex:educ + educ:exper Model 1: salary ~ sex + exper + educ sex:educ:exper Res.Df RSS Df Sum of Sq F Pr(>F)
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