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 -ifr_chisq_gof_test(hsb, race, c(20, 20, 20 , 140))
# basic example -ifr_chisq_gof_test(hsb, race, c(20, 20, 20 , 140))
## Test Statistics -## ----------------------- -## Chi-Square 5.0286 -## DF 3 -## Pr > Chi Sq 0.1697 -## Sample Size 200 -## -## Variable: race -## ----------------------------------------------------------------- -## Category Observed Expected % Deviation Std. Residuals -## ----------------------------------------------------------------- -## 1 24 20 20.00 0.89 -## 2 11 20 -45.00 -2.01 -## 3 20 20 0.00 0.00 -## 4 145 140 3.57 0.42 -## -----------------------------------------------------------------
# basic example +ifr_chisq_gof_test(hsb, race, c(20, 20, 20 , 140))
## Test Statistics +## ----------------------- +## Chi-Square 5.0286 +## DF 3 +## Pr > Chi Sq 0.1697 +## Sample Size 200 +## +## Variable: race +## ----------------------------------------------------------------- +## Category Observed Expected % Deviation Std. Residuals +## ----------------------------------------------------------------- +## 1 24 20 20.00 0.89 +## 2 11 20 -45.00 -2.01 +## 3 20 20 0.00 0.00 +## 4 145 140 3.57 0.42 +## -----------------------------------------------------------------
-# using continuity correction -ifr_chisq_gof_test(hsb, race, c(20, 20, 20 , 140), correct = TRUE)
# using continuity correction -ifr_chisq_gof_test(hsb, race, c(20, 20, 20 , 140), correct = TRUE)
## Test Statistics -## ----------------------- -## Chi-Square 4.3821 -## DF 3 -## Pr > Chi Sq 0.2231 -## Sample Size 200 -## -## Variable: race -## ----------------------------------------------------------------- -## Category Observed Expected % Deviation Std. Residuals -## ----------------------------------------------------------------- -## 1 24 20 17.50 0.78 -## 2 11 20 -47.50 -2.12 -## 3 20 20 -2.50 -0.11 -## 4 145 140 3.21 0.38 -## -----------------------------------------------------------------
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).
# using continuity correction +ifr_chisq_gof_test(hsb, race, c(20, 20, 20 , 140), correct = TRUE)
## Test Statistics +## ----------------------- +## Chi-Square 4.3821 +## DF 3 +## Pr > Chi Sq 0.2231 +## Sample Size 200 +## +## Variable: race +## ----------------------------------------------------------------- +## Category Observed Expected % Deviation Std. Residuals +## ----------------------------------------------------------------- +## 1 24 20 17.50 0.78 +## 2 11 20 -47.50 -2.12 +## 3 20 20 -2.50 -0.11 +## 4 145 140 3.21 0.38 +## -----------------------------------------------------------------
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).
-ifr_chisq_assoc_test(hsb, female, schtyp)
ifr_chisq_assoc_test(hsb, female, schtyp)
## Chi Square Statistics -## -## Statistics DF Value Prob -## ---------------------------------------------------- -## Chi-Square 1 0.0470 0.8284 -## Likelihood Ratio Chi-Square 1 0.0471 0.8282 -## Continuity Adj. Chi-Square 1 0.0005 0.9822 -## Mantel-Haenszel Chi-Square 1 0.0468 0.8287 -## Phi Coefficient 0.0153 -## Contingency Coefficient 0.0153 -## 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).
## Chi Square Statistics +## +## Statistics DF Value Prob +## ---------------------------------------------------- +## Chi-Square 1 0.0470 0.8284 +## Likelihood Ratio Chi-Square 1 0.0471 0.8282 +## Continuity Adj. Chi-Square 1 0.0005 0.9822 +## Mantel-Haenszel Chi-Square 1 0.0468 0.8287 +## Phi Coefficient 0.0153 +## Contingency Coefficient 0.0153 +## 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).
-ifr_chisq_assoc_test(hsb, schtyp, ses)
ifr_chisq_assoc_test(hsb, schtyp, ses)
## Chi Square Statistics -## -## Statistics DF Value Prob -## ---------------------------------------------------- -## Chi-Square 2 6.3342 0.0421 -## Likelihood Ratio Chi-Square 2 7.9060 0.0192 -## Phi Coefficient 0.1780 -## Contingency Coefficient 0.1752 -## Cramer's V 0.1780 -## ----------------------------------------------------
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.
## Chi Square Statistics +## +## Statistics DF Value Prob +## ---------------------------------------------------- +## Chi-Square 2 6.3342 0.0421 +## Likelihood Ratio Chi-Square 2 7.9060 0.0192 +## Phi Coefficient 0.1780 +## Contingency Coefficient 0.1752 +## Cramer's V 0.1780 +## ----------------------------------------------------
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 -ifr_levene_test(hsb, read, group_var = race)
# Using Grouping Variable -ifr_levene_test(hsb, read, group_var = race)
## Summary Statistics -## Levels Frequency Mean Std. Dev -## ----------------------------------------- -## 1 24 46.67 10.24 -## 2 11 51.91 7.66 -## 3 20 46.8 7.12 -## 4 145 53.92 10.28 -## ----------------------------------------- -## Total 200 52.23 10.25 -## ----------------------------------------- -## -## Test Statistics -## ------------------------------------------------------------------------- -## Statistic Num DF Den DF F Pr > F -## ------------------------------------------------------------------------- -## Brown and Forsythe 3 196 3.44 0.0179 -## Levene 3 196 3.4792 0.017 -## Brown and Forsythe (Trimmed Mean) 3 196 3.3936 0.019 -## -------------------------------------------------------------------------
Using the hsb data, test whether variance is equal for reading, writing and social studies scores.
# Using Grouping Variable +ifr_levene_test(hsb, read, group_var = race)
## Summary Statistics +## Levels Frequency Mean Std. Dev +## ----------------------------------------- +## 1 24 46.67 10.24 +## 2 11 51.91 7.66 +## 3 20 46.8 7.12 +## 4 145 53.92 10.28 +## ----------------------------------------- +## Total 200 52.23 10.25 +## ----------------------------------------- +## +## Test Statistics +## ------------------------------------------------------------------------- +## Statistic Num DF Den DF F Pr > F +## ------------------------------------------------------------------------- +## Brown and Forsythe 3 196 3.44 0.0179 +## Levene 3 196 3.4792 0.017 +## Brown and Forsythe (Trimmed Mean) 3 196 3.3936 0.019 +## -------------------------------------------------------------------------
Using the hsb data, test whether variance is equal for reading, +writing and social studies scores.
-# Using Variables -ifr_levene_test(hsb, read, write, socst)
# Using Variables -ifr_levene_test(hsb, read, write, socst)
## Summary Statistics -## Levels Frequency Mean Std. Dev -## ----------------------------------------- -## 0 200 52.23 10.25 -## 1 200 52.77 9.48 -## 2 200 52.41 10.74 -## ----------------------------------------- -## Total 600 52.47 10.15 -## ----------------------------------------- -## -## Test Statistics -## ------------------------------------------------------------------------- -## Statistic Num DF Den DF F Pr > F -## ------------------------------------------------------------------------- -## Brown and Forsythe 2 597 1.1683 0.3116 -## Levene 2 597 1.3803 0.2523 -## Brown and Forsythe (Trimmed Mean) 2 597 1.3258 0.2664 -## -------------------------------------------------------------------------
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.
# Using Variables +ifr_levene_test(hsb, read, write, socst)
## Summary Statistics +## Levels Frequency Mean Std. Dev +## ----------------------------------------- +## 0 200 52.23 10.25 +## 1 200 52.77 9.48 +## 2 200 52.41 10.74 +## ----------------------------------------- +## Total 600 52.47 10.15 +## ----------------------------------------- +## +## Test Statistics +## ------------------------------------------------------------------------- +## Statistic Num DF Den DF F Pr > F +## ------------------------------------------------------------------------- +## Brown and Forsythe 2 597 1.1683 0.3116 +## Levene 2 597 1.3803 0.2523 +## Brown and Forsythe (Trimmed Mean) 2 597 1.3258 0.2664 +## -------------------------------------------------------------------------
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.
-ifr_cochran_qtest(exam, exam1, exam2, exam3)
ifr_cochran_qtest(exam, exam1, exam2, exam3)
## Test Statistics -## ---------------------- -## N 15 -## Cochran's Q 4.75 -## df 2 -## p value 0.093 -## ----------------------
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:
## Test Statistics +## ---------------------- +## N 15 +## Cochran's Q 4.75 +## df 2 +## p value 0.093 +## ----------------------
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.
Using the hsb data, test if the proportion of students in himath and +hiread group is equal.
-hb <- hsb -hb$himath <- ifelse(hsb$math > 60, 1, 0) -hb$hiread <- ifelse(hsb$read > 60, 1, 0) -ifr_mcnemar_test(hb, himath, hiread)
hb <- hsb -hb$himath <- ifelse(hsb$math > 60, 1, 0) -hb$hiread <- ifelse(hsb$read > 60, 1, 0) -ifr_mcnemar_test(hb, himath, hiread)
## Controls -## --------------------------------- -## Cases 0 1 Total -## --------------------------------- -## 0 135 21 156 -## 1 18 26 44 -## --------------------------------- -## Total 153 47 200 -## --------------------------------- -## -## McNemar's Test -## ---------------------------- -## McNemar's chi2 0.2308 -## DF 1 -## Pr > chi2 0.631 -## Exact Pr >= chi2 0.7493 -## ---------------------------- -## -## Kappa Coefficient -## -------------------------------- -## Kappa 0.4454 -## ASE 0.075 -## 95% Lower Conf Limit 0.2984 -## 95% Upper Conf Limit 0.5923 -## -------------------------------- -## -## Proportion With Factor -## ---------------------- -## cases 0.78 -## controls 0.765 -## ratio 1.0196 -## odds ratio 1.1667 -## ----------------------
hb <- hsb +hb$himath <- ifelse(hsb$math > 60, 1, 0) +hb$hiread <- ifelse(hsb$read > 60, 1, 0) +ifr_mcnemar_test(hb, himath, hiread)
## Controls +## --------------------------------- +## Cases 0 1 Total +## --------------------------------- +## 0 135 21 156 +## 1 18 26 44 +## --------------------------------- +## Total 153 47 200 +## --------------------------------- +## +## McNemar's Test +## ---------------------------- +## McNemar's chi2 0.2308 +## DF 1 +## Pr > chi2 0.631 +## Exact Pr >= chi2 0.7493 +## ---------------------------- +## +## Kappa Coefficient +## -------------------------------- +## Kappa 0.4454 +## ASE 0.075 +## 95% Lower Conf Limit 0.2984 +## 95% Upper Conf Limit 0.5923 +## -------------------------------- +## +## Proportion With Factor +## ---------------------- +## cases 0.78 +## controls 0.765 +## ratio 1.0196 +## odds ratio 1.1667 +## ----------------------
-himath <- ifelse(hsb$math > 60, 1, 0) -hiread <- ifelse(hsb$read > 60, 1, 0) -ifr_mcnemar_test(table(himath, hiread))
himath <- ifelse(hsb$math > 60, 1, 0) -hiread <- ifelse(hsb$read > 60, 1, 0) -ifr_mcnemar_test(table(himath, hiread))
himath <- ifelse(hsb$math > 60, 1, 0) +hiread <- ifelse(hsb$read > 60, 1, 0) +ifr_mcnemar_test(table(himath, hiread))
Perform the above test using matrix as input.
-ifr_mcnemar_test(matrix(c(135, 18, 21, 26), nrow = 2))
ifr_mcnemar_test(matrix(c(135, 18, 21, 26), nrow = 2))
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.
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.
-# basic example -ifr_runs_test(hsb, read)
# basic example -ifr_runs_test(hsb, read)
## Runs Test -## Total Cases: 200 -## Test Value : 50 -## Cases < Test Value: 101 -## Cases > Test Value: 99 -## Number of Runs: 95 -## Expected Runs: 100.99 -## Variance (Runs): 49.73874 -## z Statistic: -0.8493358 -## p-value: 0.3956945
# basic example +ifr_runs_test(hsb, read)
## Runs Test +## Total Cases: 200 +## Test Value : 50 +## Cases < Test Value: 101 +## Cases > Test Value: 99 +## Number of Runs: 95 +## Expected Runs: 100.99 +## Variance (Runs): 49.73874 +## z Statistic: -0.8493358 +## p-value: 0.3956945
-# drop values equal to threshold -ifr_runs_test(hsb, read, drop = TRUE)
# drop values equal to threshold -ifr_runs_test(hsb, read, drop = TRUE)
## Runs Test -## Total Cases: 200 -## Test Value : 50 -## Cases < Test Value: 83 -## Cases > Test Value: 99 -## Number of Runs: 89 -## Expected Runs: 91.2967 -## Variance (Runs): 44.54805 -## z Statistic: -0.3441046 -## p-value: 0.7307676
# drop values equal to threshold +ifr_runs_test(hsb, read, drop = TRUE)
## Runs Test +## Total Cases: 200 +## Test Value : 50 +## Cases < Test Value: 83 +## Cases > Test Value: 99 +## Number of Runs: 89 +## Expected Runs: 91.2967 +## Variance (Runs): 44.54805 +## z Statistic: -0.3441046 +## p-value: 0.7307676
-# recode data in binary format -ifr_runs_test(hsb, read, split = TRUE)
# recode data in binary format -ifr_runs_test(hsb, read, split = TRUE)
# recode data in binary format +ifr_runs_test(hsb, read, split = TRUE)
-# use mean as threshold -ifr_runs_test(hsb, read, mean = TRUE)
# use mean as threshold -ifr_runs_test(hsb, read, mean = TRUE)
## Runs Test -## Total Cases: 200 -## Test Value : 52.23 -## Cases < Test Value: 115 -## Cases > Test Value: 85 -## Number of Runs: 93 -## Expected Runs: 98.75 -## Variance (Runs): 47.52418 -## z Statistic: -0.8340854 -## p-value: 0.4042329
# use mean as threshold +ifr_runs_test(hsb, read, mean = TRUE)
## Runs Test +## Total Cases: 200 +## Test Value : 52.23 +## Cases < Test Value: 115 +## Cases > Test Value: 85 +## Number of Runs: 93 +## Expected Runs: 98.75 +## Variance (Runs): 47.52418 +## z Statistic: -0.8340854 +## p-value: 0.4042329
-# threshold to be used for counting runs -ifr_runs_test(hsb, read, threshold = 0)
# threshold to be used for counting runs -ifr_runs_test(hsb, read, threshold = 0)
## Runs Test -## Total Cases: 200 -## Test Value : 0 -## Cases < Test Value: 0 -## Cases > Test Value: 200 -## Number of Runs: 1 -## Expected Runs: 1 -## Variance (Runs): 0 -## z Statistic: NaN -## p-value: NaN
# threshold to be used for counting runs +ifr_runs_test(hsb, read, threshold = 0)
## Runs Test +## Total Cases: 200 +## Test Value : 0 +## Cases < Test Value: 0 +## Cases > Test Value: 200 +## Number of Runs: 1 +## Expected Runs: 1 +## Variance (Runs): 0 +## z Statistic: NaN +## p-value: NaN