First read the Employee data included as part of lessR.
##
## >>> Suggestions
## Recommended binary format for data files: feather
## Create with Write(d, "your_file", format="feather")
## More details about your data, Enter: details() for d, or details(name)
##
## 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 7 ... 1 2 10
## 2 Gender character 37 0 2 M M W ... W W M
## 3 Dept character 36 1 5 ADMN SALE FINC ... 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 high ... high low high
## 6 Plan integer 37 0 3 1 1 2 ... 2 2 1
## 7 Pre integer 37 0 27 82 62 90 ... 83 59 80
## 8 Post integer 37 0 22 92 74 86 ... 90 71 87
## ------------------------------------------------------------------------------------------
As an option, also read the table of variable labels. Create the
table formatted as two columns. The first column is the variable name
and the second column is the corresponding variable label. Not all
variables need to be entered into the table. The table can be a
csv
file or an Excel file.
Currently, read the label file into the l data frame. The labels are displayed on both the text and visualization output. Each displayed label consists of the variable name juxtaposed with the corresponding label.
##
## >>> Suggestions
## Recommended binary format for data files: feather
## Create with Write(d, "your_file", format="feather")
## More details about your data, Enter: details() for d, or details(name)
##
## Data Types
## ------------------------------------------------------------
## character: Non-numeric data values
## ------------------------------------------------------------
##
## Variable Missing Unique
## Name Type Values Values Values First and last values
## ------------------------------------------------------------------------------------------
## 1 label character 8 0 8 Time of Company Employment ... Test score on legal issues after instruction
## ------------------------------------------------------------------------------------------
Obtain the summary statistics and 95% confidence interval for a
single variable by specifying that variable with ttest()
.
Because d is the default name of the data frame that contains
the variables for analysis, the data
parameter that names
the input data frame need not be specified.
##
## Response Variable: Salary, Annual Salary (USD)
##
##
## ------ Describe ------
##
## 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.
##
##
## ------ Infer ------
##
## 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.
##
## Response Variable: Salary, Annual Salary (USD)
##
##
## ------ Describe ------
##
## 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.
##
##
## ------ Infer ------
##
## 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.
##
## Response Variable: Salary, Annual Salary (USD)
##
##
## ------ Describe ------
##
## Salary: n = 37, mean = 73795.56, sd = 21799.53
##
##
## ------ Infer ------
##
## 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
Full analysis with ttest()
function, abbreviated as
tt()
, with formula mode.
##
## Compare Salary across Gender with levels M and W
## Response Variable: Salary, Annual Salary (USD)
## Grouping Variable: Gender, Man or Woman
##
##
## ------ Describe ------
##
## Salary for Gender M: n.miss = 0, n = 18, mean = 81147.458, sd = 23128.436
## Salary for Gender W: 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 W Shapiro-Wilk normality test: W = 0.828, p-value = 0.003
##
## Null hypothesis is equal variances of Salary, 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-value = 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 W: 11630.959
Brief version of the output contains just the basics.
##
## Compare Salary across Gender with levels M and W
## Response Variable: Salary, Annual Salary (USD)
## Grouping Variable: Gender, Man or Woman
##
##
## --- Describe ---
##
## Salary for Gender M: n.miss = 0, n = 18, mean = 81147.458, sd = 23128.436
## Salary for Gender W: 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-value = 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
##
## Compare Y across X with levels Group2 and Group1
## Response Variable: Y, Y
## Grouping Variable: X,
##
##
## --- 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-value = 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
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 – A or B –and the level of tension – L, M, or H.
Because warpbreaks is not the default data frame, specify
with the data
parameter (or set d equal to
warpbreaks).
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, Tukey’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.
##
## 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
##
##
## ANOVA
##
## 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: 0.220
## R Sq Adjusted: 0.190
## Omega Squared: 0.187
##
##
## Cohen's f: 0.479
##
##
## TUKEY MULTIPLE COMPARISONS OF MEANS
##
## Family-wise Confidence Level: 0.95
## -------------------------------
## 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: 95% family-wise confidence level
## Plot 2: Scatterplot with Cell Means
## ----------------------------------------
The brief version forgoes the multiple comparisons and the residuals.
##
## 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
##
##
## ANOVA
##
## 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: 0.220
## R Sq Adjusted: 0.190
## Omega Squared: 0.187
##
##
## Cohen's f: 0.479
##
##
## TUKEY MULTIPLE COMPARISONS OF MEANS
##
##
## RESIDUALS
Specify the second independent variable preceded by a *
sign. The plot of the cell means is generated automatically.
##
## 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
##
## Two-way Between Groups ANOVA
##
##
## DESCRIPTIVE STATISTICS
##
##
## Equal cell sizes, so balanced design
##
##
## tension
## wool L M H
## A 9 9 9
## B 9 9 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
##
## NA
##
##
##
## tension
## wool L M H
## A 18.10 8.66 10.27
## B 9.86 9.43 4.89
##
##
## ANOVA
##
##
## 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: 0.217
## Partial Omega Squared for wool: 0.049
## Partial Omega Squared for tension & wool: 0.106
##
## Cohen's f for tension: 0.527
## Cohen's f for wool: 0.226
## Cohen's f for tension_&_wool: 0.344
##
##
## TUKEY MULTIPLE COMPARISONS OF MEANS
##
## Family-wise Confidence Level: 0.95
##
## 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
## 2 L A 30.00 44.56 -14.56 -1.41
## 23 H A 10.00 24.56 -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.
## tension wool breaks_n breaks_na breaks_mean
## 1 L A 9 0 44.556
## 2 M A 9 0 24.000
## 3 H A 9 0 24.556
## 4 L B 9 0 28.222
## 5 M B 9 0 28.778
## 6 H B 9 0 18.778
Store the aggregated data in the data frame named dm, so
explicitly identify with the data
parameter. The computed
mean of breaks variable in the dm data frame from the
previous call to pivot()
is named mean by
default.
## breaks_mean
## - by levels of -
## tension
##
## n miss mean sd min mdn max
## L 2 0 36.3890 11.5499 28.2220 36.3890 44.5560
## M 2 0 26.3890 3.3786 24.0000 26.3890 28.7780
## H 2 0 21.6670 4.0857 18.7780 21.6670 24.5560
##
## Some Parameter values (can be manually set)
## -------------------------------------------------------
## size: 1.60 size of plotted points
## jitter_y: 0.00 random vertical movement of points
## jitter_x: 0.00 random horizontal movement of points
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 datavarying
: Identify the wide-form variables, which occur
over time, to be gathered into a single variable in long-formattimevar
: Name the corresponding treatment (factor)
variable in the created long-formv.names
: Name the response variable in the created
long-formThere 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 individually list each
variable.
## [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.
## 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.
##
## 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
##
## 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
##
## NA
##
##
##
## ANOVA
##
##
## 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: 0.379
## Partial Intraclass Correlation for Person: 0.785
##
## Cohen's f for Supplement: 0.782
## Cohen's f for Person: 1.911
##
##
## TUKEY MULTIPLE COMPARISONS OF MEANS
##
## Family-wise Confidence Level: 0.95
##
## 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
## 8 2 p1 4 2.75 1.25 1.60
## 11 2 p4 3 4.25 -1.25 -1.60
## 9 2 p2 5 3.75 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
## 12 2 p5 1 1.50 -0.50 -0.64
## 13 2 p6 5 5.50 -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: Data Values
## Plot 2: Fitted Values
## ---------------------
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