# Mean Inference

library("lessR")

First read the Employee data included as part of lessR.

d <- Read("Employee")
##
## >>> Suggestions
##
## Data Types
## ------------------------------------------------------------
## character: Non-numeric data values
## integer: Numeric data values, integers only
## double: Numeric data values with decimal digits
## ------------------------------------------------------------
##
##     Variable                  Missing  Unique
##         Name     Type  Values  Values  Values   First and last values
## ------------------------------------------------------------------------------------------
##  1     Years   integer     36       1      16   7  NA  15 ... 1  2  10
##  2    Gender character     37       0       2   M  M  M ... F  F  M
##  3      Dept character     36       1       5   ADMN  SALE  SALE ... MKTG  SALE  FINC
##  4    Salary    double     37       0      37   53788.26  94494.58 ... 56508.32  57562.36
##  5    JobSat character     35       2       3   med  low  low ... high  low  high
##  6      Plan   integer     37       0       3   1  1  3 ... 2  2  1
##  7       Pre   integer     37       0      27   82  62  96 ... 83  59  80
##  8      Post   integer     37       0      22   92  74  97 ... 90  71  87
## ------------------------------------------------------------------------------------------

## One-Sample t-test

Obtain the summary statistics and 95% confidence interval for a single variable by specifying that variable with ttest().

ttest(Salary)
##
##
## ------ Description ------
##
## Salary:  n.miss = 0,  n = 37,   mean = 73795.557,  sd = 21799.533
##
##
## ------ Normality Assumption ------
##
## Sample mean assumed normal because n>30, so no test needed.
##
##
## ------ Inference ------
##
## t-cutoff for 95% range of variation: tcut =  2.028
## Standard Error of Mean: SE =  3583.821
##
## Margin of Error for 95% Confidence Level:  7268.326
## 95% Confidence Interval for Mean:  66527.230 to 81063.883

Add a hypothesis test to the above.

ttest(Salary, mu=52000)
##
##
## ------ Description ------
##
## Salary:  n.miss = 0,  n = 37,   mean = 73795.557,  sd = 21799.533
##
##
## ------ Normality Assumption ------
##
## Sample mean assumed normal because n>30, so no test needed.
##
##
## ------ Inference ------
##
## t-cutoff for 95% range of variation: tcut =  2.028
## Standard Error of Mean: SE =  3583.821
##
## Hypothesized Value H0: mu = 52000
## Hypothesis Test of Mean:  t-value = 6.082,  df = 36,  p-value = 0.000
##
## Margin of Error for 95% Confidence Level:  7268.326
## 95% Confidence Interval for Mean:  66527.230 to 81063.883
##
##
## ------ Effect Size ------
##
## Distance of sample mean from hypothesized:  21795.557
## Standardized Distance, Cohen's d:  1.000
##
##
## ------ Graphics Smoothing Parameter ------
##
## Density bandwidth for 12035.673
## --------------------------------------------------

Analysis of the above from summary statistics only.

ttest(n=37, m=73795.557, s=21799.533, Ynm="Salary", mu=52000)
##
##
## ------ Description ------
##
## Salary: n = 37,   mean = 73795.56,  sd = 21799.53
##
##
## ------ Inference ------
##
## t-cutoff for 95% range of variation: tcut =  2.028
## Standard Error of Mean: SE =  3583.821
##
## Hypothesized Value H0: mu = 52000
## Hypothesis Test of Mean:  t-value = 6.082,  df = 36,  p-value = 0.000
##
## Margin of Error for 95% Confidence Level:  7268.326
## 95% Confidence Interval for Mean:  66527.231 to 81063.883
##
##
## ------ Effect Size ------
##
## Distance of sample mean from hypothesized:  21795.557
## Standardized Distance, Cohen's d:  1.000

## Two-Samples t-test

### Independent Groups

Full analysis with ttest() function, abbreviated as tt(), with formula mode.

ttest(Salary ~ Gender)
##
## Compare Salary across Gender levels M and F
##
## ------ Describe ------
##
## Salary for Gender M:  n.miss = 0,  n = 18,  mean = 81147.458,  sd = 23128.436
## Salary for Gender F:  n.miss = 0,  n = 19,  mean = 66830.598,  sd = 18438.456
##
## Mean Difference of Salary:  14316.860
##
## Weighted Average Standard Deviation:   20848.636
##
##
## ------ Assumptions ------
##
## Note: These hypothesis tests can perform poorly, and the
##       t-test is typically robust to violations of assumptions.
##       Use as heuristic guides instead of interpreting literally.
##
## Null hypothesis, for each group, is a normal distribution of Salary.
## Group M  Shapiro-Wilk normality test:  W = 0.962,  p-value = 0.647
## Group F  Shapiro-Wilk normality test:  W = 0.828,  p-value = 0.003
##
## Null hypothesis is equal variances of Salary, i.e., homogeneous.
## Variance Ratio test:  F = 534924536.348/339976675.129 = 1.573,  df = 17;18,  p-value = 0.349
## Levene's test, Brown-Forsythe:  t = 1.302,  df = 35,  p-value = 0.201
##
##
## ------ Infer ------
##
## --- Assume equal population variances of Salary for each Gender
##
## t-cutoff for 95% range of variation: tcut =  2.030
## Standard Error of Mean Difference: SE =  6857.494
##
## Hypothesis Test of 0 Mean Diff:  t = 2.088,  df = 35,  p-value = 0.044
##
## Margin of Error for 95% Confidence Level:  13921.454
## 95% Confidence Interval for Mean Difference:  395.406 to 28238.314
##
##
## --- Do not assume equal population variances of Salary for each Gender
##
## t-cutoff: tcut =  2.036
## Standard Error of Mean Difference: SE =  6900.112
##
## Hypothesis Test of 0 Mean Diff:  t = 2.075,  df = 32.505, p-value = 0.046
##
## Margin of Error for 95% Confidence Level:  14046.505
## 95% Confidence Interval for Mean Difference:  270.355 to 28363.365
##
##
## ------ Effect Size ------
##
## --- Assume equal population variances of Salary for each Gender
##
## Standardized Mean Difference of Salary, Cohen's d:  0.687
##
##
## ------ Practical Importance ------
##
## Minimum Mean Difference of practical importance: mmd
## Minimum Standardized Mean Difference of practical importance: msmd
## Neither value specified, so no analysis
##
##
## ------ Graphics Smoothing Parameter ------
##
## Density bandwidth for Gender M: 14777.329
## Density bandwidth for Gender F: 11630.959

Brief version of the output contains just the basics.

tt_brief(Salary ~ Gender)
##
## Compare Salary across Gender levels M and F
##
##  --- Describe ---
##
## Salary for Gender M:  n.miss = 0,  n = 18,  mean = 81147.458,  sd = 23128.436
## Salary for Gender F:  n.miss = 0,  n = 19,  mean = 66830.598,  sd = 18438.456
##
## Mean Difference of Salary:  14316.860
## Weighted Average Standard Deviation:   20848.636
## Standardized Mean Difference of Salary: 0.687
##
##  --- Infer ---
##
## t-cutoff for 95% range of variation: tcut =  2.030
## Standard Error of Mean Difference: SE =  6857.494
##
## Hypothesis Test of 0 Mean Diff:  t = 2.088,  df = 35,  p-value = 0.044
##
## Margin of Error for 95% Confidence Level:  13921.454
## 95% Confidence Interval for Mean Difference:  395.406 to 28238.314

### Dependent Groups

tt_brief(Pre, Post)
##
## Compare Y across X levels Group2 and Group1
##
##  --- Describe ---
##
## Y for X Group2:  n.miss = 0,  n = 37,  mean = 81.000,  sd = 11.593
## Y for X Group1:  n.miss = 0,  n = 37,  mean = 78.784,  sd = 12.037
##
## Mean Difference of Y:  2.216
## Weighted Average Standard Deviation:   11.817
## Standardized Mean Difference of Y: 0.188
##
##  --- Infer ---
##
## t-cutoff for 95% range of variation: tcut =  1.993
## Standard Error of Mean Difference: SE =  2.747
##
## Hypothesis Test of 0 Mean Diff:  t = 0.807,  df = 72,  p-value = 0.423
##
## Margin of Error for 95% Confidence Level:  5.477
## 95% Confidence Interval for Mean Difference:  -3.261 to 7.693

## ANOVA

Analysis of variance applies to the inferential analysis of means across groups. The lessR function ANOVA(), abbreviated av(), provides this analysis, based on the base R function aov().

The data for these examples is the warpbreaks data set included with the R datasets package. The data are from a weaving device called a loom for a fixed length of yarn. The response variable is the number of times the yarn broke during the weaving. Independent variables are the type of wool â€“ A or B â€“and the level of tension â€“ L, M, or H.

Because warpbreaks is not the default data frame, specify with the data parameter (or set d equal to warpbreaks).

### One-way Independent Groups

First, for illustrative purposes, ignore the type of wool and only examine the impact of tension on breaks.

The output includes descriptive statistics, ANOVA table, effect size indices, Tukeyâ€™s multiple comparisons of means, and residuals, as well as the scatterplot of the response variable with the levels of the independent variable, and a visualization of the mean comparisons.

ANOVA(breaks ~ tension, data=warpbreaks)

##   BACKGROUND
##
## Response Variable: breaks
##
## Factor Variable: tension
##   Levels: L M H
##
## Number of cases (rows) of data:  54
## Number of cases retained for analysis:  54
##
##
##   DESCRIPTIVE STATISTICS
##
##     n    mean      sd     min     max
## L  18   36.39   16.45   14.00   70.00
## M  18   26.39    9.12   12.00   42.00
## H  18   21.67    8.35   10.00   43.00
##
## Grand Mean: 28.148
##
##
##   BASIC ANALYSIS
##
##              df    Sum Sq   Mean Sq   F-value   p-value
## tension       2   2034.26   1017.13      7.21    0.0018
## Residuals    51   7198.56    141.15
##
##
## R Squared: 0.22
## Omega Squared: 0.19
##
## Cohen's f: 0.48
##
##
##   TUKEY MULTIPLE COMPARISONS OF MEANS
##
## Family-wise Confidence Level:
## -------------------------------
##         diff    lwr   upr p adj
##   M-L -10.00 -19.56 -0.44  0.04
##   H-L -14.72 -24.28 -5.16  0.00
##   H-M  -4.72 -14.28  4.84  0.46
##
##
##   RESIDUALS
##
## Fitted Values, Residuals, Standardized Residuals
##    [sorted by Standardized Residuals, ignoring + or - sign]
##    [res_rows = 20, out of 54 cases (rows) of data, or res_rows="all"]
## -------------------------------------------
##      tension breaks fitted residual z-resid
##    5       L  70.00  36.39    33.61    2.91
##    9       L  67.00  36.39    30.61    2.65
##   29       L  14.00  36.39   -22.39   -1.94
##   24       H  43.00  21.67    21.33    1.85
##    3       L  54.00  36.39    17.61    1.53
##   31       L  19.00  36.39   -17.39   -1.51
##   35       L  20.00  36.39   -16.39   -1.42
##   37       M  42.00  26.39    15.61    1.35
##    6       L  52.00  36.39    15.61    1.35
##    7       L  51.00  36.39    14.61    1.27
##   14       M  12.00  26.39   -14.39   -1.25
##   19       H  36.00  21.67    14.33    1.24
##   41       M  39.00  26.39    12.61    1.09
##   44       M  39.00  26.39    12.61    1.09
##   23       H  10.00  21.67   -11.67   -1.01
##    4       L  25.00  36.39   -11.39   -0.99
##    8       L  26.00  36.39   -10.39   -0.90
##   40       M  16.00  26.39   -10.39   -0.90
##    1       L  26.00  36.39   -10.39   -0.90
##   18       M  36.00  26.39     9.61    0.83
##
##
## ----------------------------------------
## Plot 1: Scatterplot with Cell Means
## Plot 2: 95% family-wise confidence level
## ----------------------------------------

The brief version forgoes the multiple comparisons and the residuals.

av_brief(breaks ~ tension, data=warpbreaks)

##   BACKGROUND
##
## Response Variable: breaks
##
## Factor Variable: tension
##   Levels: L M H
##
## Number of cases (rows) of data:  54
## Number of cases retained for analysis:  54
##
##
##   DESCRIPTIVE STATISTICS
##
##     n    mean      sd     min     max
## L  18   36.39   16.45   14.00   70.00
## M  18   26.39    9.12   12.00   42.00
## H  18   21.67    8.35   10.00   43.00
##
## Grand Mean: 28.148
##
##
##   BASIC ANALYSIS
##
##              df    Sum Sq   Mean Sq   F-value   p-value
## tension       2   2034.26   1017.13      7.21    0.0018
## Residuals    51   7198.56    141.15
##
##
## R Squared: 0.22
## Omega Squared: 0.19
##
## Cohen's f: 0.48
##
##
##   TUKEY MULTIPLE COMPARISONS OF MEANS
##
##   RESIDUALS

### Two-way Independent Groups

Specify the second independent variable preceded by a * sign. The plot of the cell means is generated automatically.

ANOVA(breaks ~ tension * wool, data=warpbreaks)

##   BACKGROUND
##
## Response Variable: breaks
##
## Factor Variable 1: tension
##   Levels: L M H
##
## Factor Variable 2: wool
##   Levels: A B
##
## Number of cases (rows) of data:  54
## Number of cases retained for analysis:  54
##
## The design is balanced
##
## Two-way Between Groups ANOVA
##
##
##   DESCRIPTIVE STATISTICS
##
## Cell Sample Size: 9
##
##
##       tension
##  wool     L     M     H
##     A 44.56 24.00 24.56
##     B 28.22 28.78 18.78
##
##
## tension
## ---------------------
##         L     M     H
##   1 36.39 26.39 21.67
##
## wool
## ---------------
##         A     B
##   1 31.04 25.26
##
##
## 28.148
##
##
##       tension
##  wool     L    M     H
##     A 18.10 8.66 10.27
##     B  9.86 9.43  4.89
##
##
##   BASIC ANALYSIS
##
##              df    Sum Sq   Mean Sq   F-value   p-value
##      tension  2   2034.26   1017.13      8.50    0.0007
##         wool  1    450.67    450.67      3.77    0.0582
## tension:wool  2   1002.78    501.39      4.19    0.0210
##    Residuals 48   5745.11    119.69
##
##
## Partial Omega Squared for tension: 0.22
## Partial Omega Squared for wool: 0.05
## Partial Omega Squared for tension & wool: 0.11
##
## Cohen's f for tension: 0.53
## Cohen's f for wool: 0.23
## Cohen's f for tension_&_wool: 0.34
##
##
##   TUKEY MULTIPLE COMPARISONS OF MEANS
##
## Family-wise Confidence Level:
##
## Factor: tension
## -------------------------------
##         diff    lwr   upr p adj
##   M-L -10.00 -18.82 -1.18  0.02
##   H-L -14.72 -23.54 -5.90  0.00
##   H-M  -4.72 -13.54  4.10  0.40
##
## Factor: wool
## -----------------------------
##        diff    lwr  upr p adj
##   B-A -5.78 -11.76 0.21  0.06
##
## Cell Means
## ------------------------------------
##             diff    lwr    upr p adj
##   M:A-L:A -20.56 -35.86  -5.25  0.00
##   H:A-L:A -20.00 -35.31  -4.69  0.00
##   L:B-L:A -16.33 -31.64  -1.03  0.03
##   M:B-L:A -15.78 -31.08  -0.47  0.04
##   H:B-L:A -25.78 -41.08 -10.47  0.00
##   H:A-M:A   0.56 -14.75  15.86  1.00
##   L:B-M:A   4.22 -11.08  19.53  0.96
##   M:B-M:A   4.78 -10.53  20.08  0.94
##   H:B-M:A  -5.22 -20.53  10.08  0.91
##   L:B-H:A   3.67 -11.64  18.97  0.98
##   M:B-H:A   4.22 -11.08  19.53  0.96
##   H:B-H:A  -5.78 -21.08   9.53  0.87
##   M:B-L:B   0.56 -14.75  15.86  1.00
##   H:B-L:B  -9.44 -24.75   5.86  0.46
##   H:B-M:B -10.00 -25.31   5.31  0.39
##
##
##   RESIDUALS
##
## Fitted Values, Residuals, Standardized Residuals
##    [sorted by Standardized Residuals, ignoring + or - sign]
##    [res_rows = 20, out of 54 cases (rows) of data, or res_rows="all"]
## ------------------------------------------------
##      tension wool breaks fitted residual z-resid
##    5       L    A  70.00  44.56    25.44    2.47
##    9       L    A  67.00  44.56    22.44    2.18
##    4       L    A  25.00  44.56   -19.56   -1.90
##    8       L    A  26.00  44.56   -18.56   -1.80
##    1       L    A  26.00  44.56   -18.56   -1.80
##   24       H    A  43.00  24.56    18.44    1.79
##   36       L    B  44.00  28.22    15.78    1.53
##   23       H    A  10.00  24.56   -14.56   -1.41
##    2       L    A  30.00  44.56   -14.56   -1.41
##   29       L    B  14.00  28.22   -14.22   -1.38
##   37       M    B  42.00  28.78    13.22    1.28
##   34       L    B  41.00  28.22    12.78    1.24
##   40       M    B  16.00  28.78   -12.78   -1.24
##   14       M    A  12.00  24.00   -12.00   -1.16
##   18       M    A  36.00  24.00    12.00    1.16
##   19       H    A  36.00  24.56    11.44    1.11
##   16       M    A  35.00  24.00    11.00    1.07
##   41       M    B  39.00  28.78    10.22    0.99
##   44       M    B  39.00  28.78    10.22    0.99
##   39       M    B  19.00  28.78    -9.78   -0.95

Can also obtain the cell mean plot directly from the means. Here use lessR pivot() to compute the cell means of breaks across tension and wool.

data(warpbreaks)
dm <- pivot(warpbreaks, mean, breaks, c(tension, wool))
Plot(tension, breaks, by=wool, segments=TRUE, size=2, data=dm, main="Cell Means")
##
## >>> Note
## The integrated Violin/Box/Scatterplot (VBS) for breaks
## at each level of tension is only obtained if the categorical
## variable is the variable listed second, that is, the y-variable.
##
## This ordering with tension listed first yields the
## scatterplot and the associated means, but no VBS plot.

## >>> Suggestions
## Plot(tension, breaks, data=dm, by=wool, size=2, segments=TRUE, main="Cell Means", means=FALSE)  # do not plot means
## Plot(tension, breaks, data=dm, by=wool, size=2, segments=TRUE, main="Cell Means", stat="mean")  # only plot means
## ANOVA(breaks ~ tension)  # inferential analysis
##
##
## breaks
##   - by levels of -
## tension
##
##     n   miss            mean              sd             min             mdn             max
## L   2      0       36.388889       11.549411       28.222222       36.388889       44.555556
## M   2      0       26.388889        3.378399       24.000000       26.388889       28.777778
## H   2      0       21.666667        4.085506       18.777778       21.666667       24.555556

### Randomized Block Design

The randomized block design has a treatment variable, usually administered over time, and a blocking variable. The values of the treatment variable are measured across each instance of the blocking variable. In this example, repetitions are measured across four different workout sessions. The person takes one of four supplements before each session. Person is the blocking variable, and Supplement is the treatment variable. Repetitions is the response variable.

The data are presented in a wide-form data table, a single row for each person.

d <- read.csv(header=TRUE, text="
Person,sup1,sup2,sup3,sup4
p1,2,4,4,3
p2,2,5,4,6
p3,8,6,7,9
p4,4,3,5,7
p5,2,1,2,3
p6,5,5,6,8
p7,2,3,2,4")

The ANOVA, however, requires data to be in long-form. Reshape data from wide form to long form with base R reshape() according to the following parameters. With each parameter, either identify existing variables in the given wide-form data, or name newly created variables in the long-form. This R function refers to a time-variable, which in the context of ANOVA is the treatment variable, of which the values occur over time: first treatment, second treatment, etc.

The reshaping from a wide-form to a long-form data table creates two new variables: the variable whose values are collected over time, here the blocking variable, Supplement, and the response variable, here Reps.

• idvar: Identify the existing blocking (within) variable in the wide-form data
• varying: Identify the wide-form variables, which occur over time, to be gathered into a single variable in long-format
• timevar: Name the corresponding treatment (factor) variable in the created long-form
• v.names: Name the response variable in the created long-form

There are many ways to identify the names of the wide-form variables to be gathered into a single time-oriented long-form variable. The most general is to specify a vector of the names, here

c("sup1", "sup2" "sup3", "sup4")

In this example use the lessR to function to create that vector without needed to individual list each variable.

to("sup", 4)
## [1] "sup1" "sup2" "sup3" "sup4"
d <- reshape(d, direction="long",
idvar="Person", varying=list(to("sup", 4)),
timevar="Supplement", v.names="Reps")

Do not need the row names, so remove before displaying new long-form data.

row.names(d) <- NULL
d[1:10,]
##    Person Supplement Reps
## 1      p1          1    2
## 2      p2          1    2
## 3      p3          1    8
## 4      p4          1    4
## 5      p5          1    2
## 6      p6          1    5
## 7      p7          1    2
## 8      p1          2    4
## 9      p2          2    5
## 10     p3          2    6

To run the ANOVA, specify the blocking variable preceded by a + sign.

ANOVA(Reps ~ Supplement + Person)

##   BACKGROUND
##
## Response Variable: Reps
##
## Factor Variable 1: Supplement
##   Levels: 1 2 3 4
##
## Factor Variable 2: Person
##   Levels: p1 p2 p3 p4 p5 p6 p7
##
## Number of cases (rows) of data:  28
## Number of cases retained for analysis:  28
##
## The design is balanced
##
## Randomized Blocks ANOVA
##   Factor of Interest:  Supplement
##   Blocking Factor:     Person
##
## Note: For the resulting F statistic for Supplement to be distributed as F,
##       the population covariances of Reps must be spherical.
##
##
##   DESCRIPTIVE STATISTICS
##
## Supplement
## -----------------------
##       X1   X2   X3   X4
##   1 3.57 3.86 4.29 5.71
##
## Person
## --------------------------------------
##       p1   p2   p3   p4   p5   p6   p7
##   1 3.25 4.25 7.50 4.75 2.00 6.00 2.75
##
##
## 4.357
##
##
##   BASIC ANALYSIS
##
##            df    Sum Sq   Mean Sq   F-value   p-value
## Supplement  3     19.00      6.33      6.71    0.0031
##     Person  6     88.43     14.74     15.61    0.0000
##  Residuals 18     17.00      0.94
##
##
## Partial Omega Squared for Supplement: 0.38
## Partial Intraclass Correlation for Person: 0.79
##
## Cohen's f for Supplement: 0.78
## Cohen's f for Person: 1.91
##
##
##   TUKEY MULTIPLE COMPARISONS OF MEANS
##
## Family-wise Confidence Level:
##
## Factor: Supplement
## ---------------------------
##       diff   lwr  upr p adj
##   2-1 0.29 -1.18 1.75  0.95
##   3-1 0.71 -0.75 2.18  0.53
##   4-1 2.14  0.67 3.61  0.00
##   3-2 0.43 -1.04 1.90  0.84
##   4-2 1.86  0.39 3.33  0.01
##   4-3 1.43 -0.04 2.90  0.06
##
##
##   RESIDUALS
##
## Fitted Values, Residuals, Standardized Residuals
##    [sorted by Standardized Residuals, ignoring + or - sign]
##    [res_rows = 20, out of 28 cases (rows) of data, or res_rows="all"]
## ---------------------------------------------------
##      Supplement Person Reps fitted residual z-resid
##   22          4     p1    3   4.61    -1.61   -2.06
##    2          1     p2    2   3.46    -1.46   -1.88
##    3          1     p3    8   6.71     1.29    1.65
##    9          2     p2    5   3.75     1.25    1.60
##    8          2     p1    4   2.75     1.25    1.60
##   11          2     p4    3   4.25    -1.25   -1.60
##   10          2     p3    6   7.00    -1.00   -1.28
##   25          4     p4    7   6.11     0.89    1.15
##   15          3     p1    4   3.18     0.82    1.05
##    5          1     p5    2   1.21     0.79    1.01
##   14          2     p7    3   2.25     0.75    0.96
##   21          3     p7    2   2.68    -0.68   -0.87
##   27          4     p6    8   7.36     0.64    0.83
##   13          2     p6    5   5.50    -0.50   -0.64
##   12          2     p5    1   1.50    -0.50   -0.64
##    1          1     p1    2   2.46    -0.46   -0.60
##   17          3     p3    7   7.43    -0.43   -0.55
##   23          4     p2    6   5.61     0.39    0.50
##   26          4     p5    3   3.36    -0.36   -0.46
##   18          3     p4    5   4.68     0.32    0.41
##
##
## ------------------------
## Plot 1: Interaction Plot
## Plot 2: Fitted Values
## ------------------------

## Full Manual

Use the base R help() function to view the full manual for ttest() or ANOVA(). Simply enter a question mark followed by the name of the function.

?ttest
?ANOVA