Inferential statistics allows us to make generalizations about populations using data drawn from the population. We use them when it is impractical or impossible to collect data about the whole population under study and instead, we have a sample that represents the population under study and using inferential statistics technique, we make generalizations about the population from the sample. inferr builds upon the solid set of statistical tests provided in stats package by including additional data types as inputs, expanding and restructuring the test results.
The inferr package:
As of version 0.1, inferr includes a select set of parametric and non-parametric statistical tests which are listed below:
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.
## One-Sample Statistics
## ---------------------------------------------------------------------------------
## Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
## ---------------------------------------------------------------------------------
## write 200 52.775 0.6702 9.4786 51.4537 54.0969
## ---------------------------------------------------------------------------------
##
## Ho: mean(write) ~=50
##
## Ha: mean < 50 Ha: mean ~= 50 Ha: mean > 50
## t = 4.141 t = 4.141 t = 4.141
## P < t = 0.99997 P > |t| = 0.00005 P > t = 0.00003A 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.
## Paired Samples Statistics
## ----------------------------------------------------------------------------
## Variables Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
## ----------------------------------------------------------------------------
## read 200 52.23 0.7249921 10.25294 50.8 53.66
## write 200 52.775 0.6702372 9.478586 51.45 54.1
## ----------------------------------------------------------------------------
## diff 200 -0.545 0.6283822 8.886666 -1.78 0.69
## ----------------------------------------------------------------------------
##
## Paired Samples Correlations
## -------------------------------------------
## Variables Obs Correlation Sig.
## read & write 200 0.60 0
## -------------------------------------------
##
## Paired Samples Test
## -------------------
## Ho: mean(read - write) = 0
## Ha: mean(read - write) < 0
##
## ----------------------------------------
## Variables t df Sig.
## ----------------------------------------
## read - write -0.8673 199 0.193
## ----------------------------------------## Paired Samples Statistics
## ----------------------------------------------------------------------------
## Variables Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
## ----------------------------------------------------------------------------
## read 200 52.23 0.7249921 10.25294 50.8 53.66
## write 200 52.775 0.6702372 9.478586 51.45 54.1
## ----------------------------------------------------------------------------
## diff 200 -0.545 0.6283822 8.886666 -1.78 0.69
## ----------------------------------------------------------------------------
##
## Paired Samples Correlations
## -------------------------------------------
## Variables Obs Correlation Sig.
## read & write 200 0.60 0
## -------------------------------------------
##
## Ho: mean(read - write) = mean(diff) = 0
##
## Ha: mean(diff) < 0 Ha: mean(diff) ~= 0 Ha: mean(diff) > 0
## t = -0.8673 t = -0.8673 t = -0.8673
## P < t = 0.193 P > |t| = 0.387 P > t = 0.807An 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.
## Group Statistics
## -----------------------------------------------------------------------------
## Group Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
## -----------------------------------------------------------------------------
## 0 91 50.121 1.080 10.305 47.97539 52.26661
## 1 109 54.991 0.779 8.134 53.44689 56.53511
## -----------------------------------------------------------------------------
## combined 200 52.775 0.67 9.479 51.45379 54.09621
## -----------------------------------------------------------------------------
## diff 200 -4.87 1.304 9.231 -7.42623 -2.31377
## -----------------------------------------------------------------------------
##
## Independent Samples Test
## ------------------------
##
## Ho: mean(0) - mean(1) = diff = 0
##
## Ha: diff < 0 Ha: diff ~= 0 Ha: diff > 0
##
## Pooled
## ------------------------------------------------------------------------
## t = -3.7341 t = -3.7341 t = -3.7341
## P < t = 0.0001 P > |t| = 0.0002 P > t = 0.9999
##
## Satterthwaite
## ------------------------------------------------------------------------
## t = -3.6564 t = -3.6564 t = -3.6564
## P < t = 0.0002 P > |t| = 0.0003 P > t = 0.9998
##
##
## Test for Equality of Variances
## ---------------------------------------------------------------
## Variable Method Num DF Den DF F Value P > F
## ---------------------------------------------------------------
## write Folded F 90 108 1.605 0.0188
## ---------------------------------------------------------------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%.
## Test Statistics
## -------------------------
## Sample Size 200
## Exp Prop 0.5
## Obs Prop 0.545
## z 1.2728
## Pr(|Z| > |z|) 0.2031
##
## -----------------------------------------------------------------
## Category Observed Expected % Deviation Std. Residuals
## -----------------------------------------------------------------
## 0 91 100 -9.00 -0.90
## 1 109 100 9.00 0.90
## -----------------------------------------------------------------Using Calculator
## Test Statistics
## --------------------------
## Sample Size 200
## Exp Prop 0.5
## Obs Prop 0.3
## z -5.6569
## Pr(|Z| > |z|) 0
##
## -----------------------------------------------------------------
## Category Observed Expected % Deviation Std. Residuals
## -----------------------------------------------------------------
## 0 140 100 40.00 4.00
## 1 60 100 -40.00 -4.00
## -----------------------------------------------------------------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
## Test Statistics
## -----------------------------
## Total Observations 100
## z 0.403
## Pr(|Z| < |z|) 0.687
## Pr(Z < z) 0.656
## Pr(Z > z) 0.344Using the treatment2 data, test whether outcome has same proportion for male and female
## Test Statistics
## -----------------------------
## Total Observations 200
## z 0.351
## Pr(|Z| < |z|) 0.726
## Pr(Z < z) 0.637
## Pr(Z > z) 0.363Test 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.
## Test Statistics
## ------------------------------
## Total Observations 55
## z -1.514
## Pr(|Z| < |z|) 0.13
## Pr(Z < z) 0.065
## Pr(Z > z) 0.935One 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.
## One-Sample Statistics
## -----------------------------------------------------------------------------
## Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
## -----------------------------------------------------------------------------
## mpg 32 20.0906 1.0654 6.0269 3.8737 10.6527
## -----------------------------------------------------------------------------
##
## Lower Tail Test
## ---------------
## Ho: sd(mpg) >= 0.3
## Ha: sd(mpg) < 0.3
##
## Chi-Square Test for Variance
## ----------------------------------------
## Variable c DF Sig
## ----------------------------------------
## mpg 12511.635 31 1.0000
## ----------------------------------------## One-Sample Statistics
## -----------------------------------------------------------------------------
## Variable Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
## -----------------------------------------------------------------------------
## mpg 32 20.0906 1.0654 6.0269 3.8737 10.6527
## -----------------------------------------------------------------------------
##
## Ho: sd(mpg) = 0.3
##
## Ha: sd < 0.3 Ha: sd != 0.3 Ha: sd > 0.3
## c = 12511.6354 c = 12511.6354 c = 12511.6354
## Pr(C < c) = 1.0000 2 * Pr(C > c) = 0.0000 Pr(C > c) = 0.0000Two 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.
## Variance Ratio Test
## --------------------------------------------------
## Group Obs Mean Std. Err. Std. Dev.
## --------------------------------------------------
## 0 91 52.82 1.1 10.51
## 1 109 51.73 0.96 10.06
## --------------------------------------------------
## combined 200 52.23 0.72 10.25
## --------------------------------------------------
##
## Variance Ratio Test
## --------------------------------------------------
## F Num DF Den DF
## --------------------------------------------------
## 1.0913 90 108
## --------------------------------------------------
##
## Null & Alternate Hypothesis
## ----------------------------------------
## ratio = sd(0) / (1)
## Ho: ratio = 1
##
## Ha: ratio < 1 Ha: ratio > 1
## Pr(F < f) = 0.6694 Pr(F > f) = 0.3306
## ----------------------------------------Using the hsb data, compare the standard deviation of reading and writing scores.
## Variance Ratio Test
## --------------------------------------------------
## Group Obs Mean Std. Err. Std. Dev.
## --------------------------------------------------
## read 200 52.23 0.72 10.25
## write 200 52.77 0.67 9.48
## --------------------------------------------------
## combined 400 52.5 0.49 9.86
## --------------------------------------------------
##
## Variance Ratio Test
## --------------------------------------------------
## F Num DF Den DF
## --------------------------------------------------
## 1.1701 199 199
## --------------------------------------------------
##
## Null & Alternate Hypothesis
## ----------------------------------------
## ratio = sd(read) / (write)
## Ho: ratio = 1
##
## Ha: ratio < 1 Ha: ratio > 1
## Pr(F < f) = 0.8656 Pr(F > f) = 0.1344
## ----------------------------------------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.
## Binomial Test
## ---------------------------------------
## Group N Obs. Prop Exp. Prop
## ---------------------------------------
## 0 91 0.455 0.500
## 1 109 0.545 0.500
## ---------------------------------------
##
##
## Test Summary
## ----------------------------------------------
## Tail Prob p-value
## ----------------------------------------------
## Lower Pr(k <= 109) 0.910518
## Upper Pr(k >= 109) 0.114623
## ----------------------------------------------## Binomial Test
## --------------------------------------
## Group N Obs. Prop Exp. Prop
## --------------------------------------
## 0 16 0.5 0.500
## 1 16 0.5 0.500
## --------------------------------------
##
##
## Test Summary
## --------------------------------------------
## Tail Prob p-value
## --------------------------------------------
## 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.
## ANOVA
## ----------------------------------------------------------------------
## Sum of
## Squares DF Mean Square F Sig.
## ----------------------------------------------------------------------
## Between Groups 3175.698 2 1587.849 21.275 0
## Within Groups 14703.177 197 74.635
## Total 17878.875 199
## ----------------------------------------------------------------------
##
## Report
## -----------------------------------------
## Category N Mean Std. Dev.
## -----------------------------------------
## 1 45 51.333 9.398
## 2 105 56.257 7.943
## 3 50 46.760 9.319
## -----------------------------------------
##
## Number of obs = 200 R-squared = 0.1776
## Root MSE = 8.6392 Adj R-squared = 0.1693A 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.
## 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
## -----------------------------------------------------------------## 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).
## 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 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.
## 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.
## 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.
## 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.
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))## 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
## ----------------------Perform the above test using matrix as input.
## 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
## ----------------------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
## 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## 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## 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## 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## 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