These tests are described in more detail in the following sections.
A one sample t-test is used to determine whether a sample of observations comes from a population with a specific mean. The observations must be continuous, independent of each other, approximately distributed and should not contain any outliers.
Using the hsb data, test whether the average of write differs significantly from 50.
-infer_os_t_test(hsb$write, mu = 50, type = 'all')## One-Sample Statistics
## ---------------------------------------------------------------------------------
## Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
@@ -138,14 +138,14 @@
A paired (samples) t-test is used when you want to compare the means between two related groups of observations on some continuous dependent variable. In a paired sample test, each subject or entity is measured twice. It can be used to evaluate the effectiveness of training programs or treatments. If the dependent variable is dichotomous, use the McNemar test.
Using the hsb data, test whether the mean of read is equal to the mean of write.
-# Lower Tail Test
-infer_ts_paired_ttest(hsb$read, hsb$write, alternative = 'less')## Paired Samples Statistics
## ---------------------------------------------------------------------------
## Variables Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
@@ -172,8 +172,8 @@
## ---------------------------------------
## read - write -0.873 199 0.192
## ---------------------------------------
# Test all alternatives
-infer_ts_paired_ttest(hsb$read, hsb$write, alternative = 'all')## Paired Samples Statistics
## ---------------------------------------------------------------------------
## Variables Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
@@ -199,7 +199,7 @@
An independent samples t-test is used to compare the means of a normally distributed continuous dependent variable for two unrelated groups. The dependent variable must be approximately normally distributed and the cases/subjects in the two groups must be different i.e. a subject in one group cannot also be a subject of the other group. It can be used to answer whether:
Using the hsb data, test whether the mean for write is the same for males and females.
-hsb2 <- inferr::hsb
-hsb2$female <- as.factor(hsb2$female)
-infer_ts_ind_ttest(hsb2, 'female', 'write', alternative = 'all')hsb2 <- inferr::hsb
+hsb2$female <- as.factor(hsb2$female)
+infer_ts_ind_ttest(hsb2, 'female', 'write', alternative = 'all')## Group Statistics
## -----------------------------------------------------------------------------
## Group Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
@@ -252,14 +252,14 @@
One sample test of proportion compares proportion in one group to a specified population proportion.
Using hsb data, test whether the proportion of females is 50%.
-# Using Variables
-infer_os_prop_test(as.factor(hsb$female), prob = 0.5)## Test Statistics
## -------------------------
## Sample Size 200
@@ -275,8 +275,8 @@
## 1 109 100 9.00 0.90
## -----------------------------------------------------------------
Using Calculator
-# Calculator
-infer_os_prop_test(200, prob = 0.5, phat = 0.3)## Test Statistics
## --------------------------
## Sample Size 200
@@ -295,17 +295,17 @@
Two sample test of proportion performs tests on the equality of proportions using large-sample statistics. It tests that a categorical variable has the same proportion within two groups or that two variables have the same proportion.
Using the treatment data, test equality of proportion of two treatments
-# Using Variables
-infer_ts_prop_test(var1 = treatment$treatment1, var2 = treatment$treatment2, alternative = 'all')# Using Variables
+infer_ts_prop_test(var1 = treatment$treatment1, var2 = treatment$treatment2, alternative = 'all')## Test Statistics
## ------------------------
## Sample Size 50
@@ -316,10 +316,10 @@
Using the treatment2 data, test whether outcome has same proportion for male and female
-# Using Grouping Variable
-infer_ts_prop_grp(var = treatment2$outcome, group = treatment2$female, alternative = 'all')# Using Grouping Variable
+infer_ts_prop_grp(var = treatment2$outcome, group = treatment2$female, alternative = 'all')## Test Statistics
## ------------------------
## Sample Size 91
@@ -330,10 +330,10 @@
Test whether the same proportion of people from two batches will pass a review exam for a training program. In the first batch of 30 participants, 30% passed the review, whereas in the second batch of 25 participants, 50% passed the review.
-# Calculator
-infer_ts_prop_calc(n1 = 30, n2 = 25, p1 = 0.3, p2 = 0.5, alternative = 'all')## Test Statistics
## -------------------------
## Sample Size 30
@@ -346,7 +346,7 @@
One sample variance comparison test compares the standard deviation (variances) to a hypothesized value. It determines whether the standard deviation of a population is equal to a hypothesized value. It can be used to answer the following questions:
Using the mtcars data, compare the standard deviation of mpg to a hypothesized value.
-# Lower Tail Test
-infer_os_var_test(mtcars$mpg, 0.3, alternative = 'less')## One-Sample Statistics
## -----------------------------------------------------------------------------
## Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
@@ -377,8 +377,8 @@
## ----------------------------------------
## mpg 12511.436 31 1.0000
## ----------------------------------------
# Test all alternatives
-infer_os_var_test(mtcars$mpg, 0.3, alternative = 'all')## One-Sample Statistics
## -----------------------------------------------------------------------------
## Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
@@ -395,17 +395,17 @@
Two sample variance comparison tests equality of standard deviations (variances). It tests that the standard deviation of a continuous variable is same within two groups or the standard deviation of two continuous variables is equal.
Using the mtcars data, compare the standard deviation in miles per gallon for automatic and manual vehicles.
-# Using Grouping Variable
-infer_ts_var_test(mtcars$mpg, group_var = mtcars$am, alternative = 'all')# Using Grouping Variable
+infer_ts_var_test(mtcars$mpg, group_var = mtcars$am, alternative = 'all')## Variance Ratio Test
## --------------------------------------------------
## Group Obs Mean Std. Err. Std. Dev.
@@ -434,10 +434,10 @@
Using the hsb data, compare the standard deviation of reading and writing scores.
-# Using Variables
-infer_ts_var_test(hsb$read, hsb$write, alternative = 'all')## Variance Ratio Test
## --------------------------------------------------
## Group Obs Mean Std. Err. Std. Dev.
@@ -468,14 +468,14 @@
A one sample binomial test allows us to test whether the proportion of successes on a two-level categorical dependent variable significantly differs from a hypothesized value.
Using the hsb data, test whether the proportion of females and males are equal.
-# Using variables
-infer_binom_test(as.factor(hsb$female), prob = 0.5)## Binomial Test
## ---------------------------------------
## Group N Obs. Prop Exp. Prop
@@ -485,19 +485,18 @@
## ---------------------------------------
##
##
-## Test Summary
-## ---------------------------------------------
-## Tail Prob p-value
-## ---------------------------------------------
-## Lower Pr(k <= 109) 0.910518
-## Upper Pr(k >= 109) 0.114623
-## Two Pr(k <= 91 or k >= 109) 0.229247
-## ---------------------------------------------
# calculator
-infer_binom_calc(32, 16, prob = 0.5)## Binomial Test
## --------------------------------------
## Group N Obs. Prop Exp. Prop
@@ -511,22 +510,21 @@
## --------------------------------------------
## Tail Prob p-value
## --------------------------------------------
-## Lower Pr(k <= 16) 0.569975
-## Upper Pr(k >= 16) 0.569975
-## Two Pr(k <= 15 or k >= 16) 1
+## Lower Pr(k <= 16) 0.569975
+## Upper Pr(k >= 16) 0.569975
## --------------------------------------------
The one-way analysis of variance (ANOVA) is used to determine whether there are any statistically significant differences between the means of two or more independent (unrelated) groups. It tests the null hypothesis that samples in two or more groups are drawn from populations with the same mean values. It cannot tell you which specific groups were statistically significantly different from each other but only that at least two groups were different and can be used only for numerical data.
Using the hsb data, test whether the mean of write differs between the three program types.
-infer_oneway_anova(hsb, 'write', 'prog')## ANOVA
## ----------------------------------------------------------------------
## Sum of
@@ -552,15 +550,15 @@
A chi-square goodness of fit test allows us to compare the observed sample distribution with expected probability distribution. It tests whether the observed proportions for a categorical variable differ from hypothesized proportions. The proportion of cases expected in each group of categorical variable may be equal or unequal. It can be applied to any univariate distribution for which you can calculate the cumulative distribution function. It is applied to binned data and the value of the chi square test depends on how the data is binned. For the chi square approximation to be valid, the sample size must be sufficiently large.
Using the hsb data, test whether the observed proportions for race differs significantly from the hypothesized proportions.
-# basic example
-race <- as.factor(hsb$race)
-infer_chisq_gof_test(race, c(20, 20, 20 , 140))## Test Statistics
## -----------------------
## Chi-Square 5.0286
@@ -579,10 +577,10 @@
## -----------------------------------------------------------------
# using continuity correction
-race <- as.factor(hsb$race)
-infer_chisq_gof_test(race, c(20, 20, 20 , 140), correct = TRUE)# using continuity correction
+race <- as.factor(hsb$race)
+infer_chisq_gof_test(race, c(20, 20, 20 , 140), correct = TRUE)## Test Statistics
## -----------------------
## Chi-Square 4.3821
@@ -604,13 +602,13 @@
A chi-square test is used when you want to test if there is a significant relationship between two nominal (categorical) variables.
Using the hsb data, test if there is a relationship between the type of school attended (schtyp) and students’ gender (female).
-infer_chisq_assoc_test(as.factor(hsb$female), as.factor(hsb$schtyp))## Chi Square Statistics
##
## Statistics DF Value Prob
@@ -624,7 +622,7 @@
## Cramer's V 0.0153
## ----------------------------------------------------
Using the hsb data, test if there is a relationship between the type of school attended (schtyp) and students’ socio economic status (ses).
-infer_chisq_assoc_test(as.factor(hsb$schtyp), as.factor(hsb$ses))## Chi Square Statistics
##
## Statistics DF Value Prob
@@ -639,17 +637,17 @@
Levene’s test is used to determine if k samples have equal variances. It is less sensitive to departures from normality and is an alternative to Bartlett’s test. This test returns Levene’s robust test statistic and the two statistics proposed by Brown and Forsythe that replace the mean in Levene’s formula with alternative location estimators. The first alternative replaces the mean with the median and the second alternative replaces the mean with the 10% trimmed mean.
Using the hsb data, test whether variance in reading score is same across race.
-# Using Grouping Variable
-infer_levene_test(hsb$read, group_var = hsb$race)## Summary Statistics
## Levels Frequency Mean Std. Dev
## -----------------------------------------
@@ -672,10 +670,10 @@
Using the hsb data, test whether variance is equal for reading, writing and social studies scores.
-# Using Variables
-infer_levene_test(hsb$read, hsb$write, hsb$socst)## Summary Statistics
## Levels Frequency Mean Std. Dev
## -----------------------------------------
@@ -697,11 +695,11 @@
Using the hsb data, test whether variance in reading score is same for male and female students.
-# Using Linear Regression Model
-m <- lm(read ~ female, data = hsb)
-infer_levene_test(m)## Summary Statistics
## Levels Frequency Mean Std. Dev
## -----------------------------------------
@@ -722,10 +720,10 @@
Using the hsb data, test whether variance in reading score is same across school types.
-# Using Formula
-infer_levene_test(as.formula(paste0('read ~ schtyp')), hsb)## Summary Statistics
## Levels Frequency Mean Std. Dev
## -----------------------------------------
@@ -748,13 +746,13 @@
Cochran’s Q test is an extension to the McNemar test for related samples that provides a method for testing for differences between three or more matched sets of frequencies or proportions. It is a procedure for testing if the proportions of 3 or more dichotomous variables are equal in some population. These outcome variables have been measured on the same people or other statistical units.
The exam data set contains scores of 15 students for three exams (exam1, exam2, exam3). Test if three exams are equally difficult.
-infer_cochran_qtest(exam)## Test Statistics
## ----------------------
## N 15
@@ -766,7 +764,7 @@
McNemar test is a non parametric test created by Quinn McNemar and first published in Psychometrika in 1947. It is similar to a paired t test but applied to a dichotomous dependent variable. It is used to test if a statistically significant change in proportions have occurred on a dichotomous trait at two time points on the same population. It can be used to answer whether:
Using the hsb data, test if the proportion of students in himath and hiread group is equal.
-himath <- ifelse(hsb$math > 60, 1, 0)
-hiread <- ifelse(hsb$read > 60, 1, 0)
-infer_mcnemar_test(table(himath, hiread))himath <- ifelse(hsb$math > 60, 1, 0)
+hiread <- ifelse(hsb$read > 60, 1, 0)
+infer_mcnemar_test(table(himath, hiread))## Controls
## ---------------------------------
## Cases 0 1 Total
@@ -814,7 +812,7 @@
## odds ratio 1.1667
## ----------------------
Perform the above test using matrix as input.
-infer_mcnemar_test(matrix(c(135, 18, 21, 26), nrow = 2))## Controls
## ---------------------------------
## Cases 0 1 Total
@@ -852,17 +850,17 @@
Runs Test can be used to decide if a data set is from a random process. It tests whether observations of a sequence are serially independent i.e. whether they occur in a random order by counting how many runs there are above and below a threshold. A run is defined as a series of increasing values or a series of decreasing values. The number of increasing, or decreasing, values is the length of the run. By default, the median is used as the threshold. A small number of runs indicates positive serial correlation; a large number indicates negative serial correlation.
We will use runs test to check regression residuals for serial correlation.
-# linear regression
-reg <- lm(mpg ~ disp, data = mtcars)
-
-# basic example
-infer_runs_test(residuals(reg))# linear regression
+reg <- lm(mpg ~ disp, data = mtcars)
+
+# basic example
+infer_runs_test(residuals(reg))## Runs Test
## Total Cases: 32
## Test Value : -0.9630856
@@ -873,8 +871,8 @@
## Variance (Runs): 7.741935
## z Statistic: -2.156386
## p-value: 0.03105355
# drop values equal to threshold
-infer_runs_test(residuals(reg), drop = TRUE)## Runs Test
## Total Cases: 32
## Test Value : -0.9630856
@@ -885,8 +883,8 @@
## Variance (Runs): 7.741935
## z Statistic: -2.156386
## p-value: 0.03105355
# recode data in binary format
-infer_runs_test(residuals(reg), split = TRUE)## Runs Test
## Total Cases: 32
## Test Value : -0.9630856
@@ -897,8 +895,8 @@
## Variance (Runs): 7.741935
## z Statistic: -2.156386
## p-value: 0.03105355
# use mean as threshold
-infer_runs_test(residuals(reg), mean = TRUE)## Runs Test
## Total Cases: 32
## Test Value : -1.12757e-16
@@ -909,8 +907,8 @@
## Variance (Runs): 7.189642
## z Statistic: -2.027896
## p-value: 0.04257089
# threshold to be used for counting runs
-infer_runs_test(residuals(reg), threshold = 0)## Runs Test
## Total Cases: 32
## Test Value : 0
@@ -925,7 +923,7 @@
The examples and the data set used in the vignette are borrowed from the below listed sources: