multcompare
Description
specifies additional options using one or more name-value arguments. For example, you can
specify the confidence level and the type of critical value used to determine if the means
are significantly different.m
= multcompare(___,Name=Value
)
Examples
Compare Group Means of One-Way ANOVA
Load popcorn yield data.
load popcorn.mat
The columns of the 6-by-3 matrix popcorn
contain popcorn yield observations in cups for the brands Gourmet, National, and Generic.
Convert popcorn
to a vector.
popcorn = popcorn(:);
Create a string array of values for the factor Brand
using the function repmat
.
brand = [repmat("Gourmet",6,1); repmat("National",6,1); repmat("Generic",6,1)];
Perform a one-way ANOVA to test the null hypothesis that the mean yields are the same across the three brands.
aov = anova(brand,popcorn,FactorNames="Brand")
aov = 1-way anova, constrained (Type III) sums of squares. Y ~ 1 + Brand SumOfSquares DF MeanSquares F pValue ____________ __ ___________ ____ __________ Brand 15.75 2 7.875 18.9 7.9603e-05 Error 6.25 15 0.41667 Total 22 17 Properties, Methods
The small p-value indicates that the null hypothesis can be rejected at the 99% confidence level. Therefore, the difference in mean popcorn yield is statistically significant for at least one brand. Perform Dunnett's Test to determine if the mean yields of Gourmet
and National
differ significantly from the mean yield of Generic
.
m = multcompare(aov,CriticalValueType="dunnett",ControlGroup=3)
m=2×6 table
Group1 Group2 MeanDifference MeanDifferenceLower MeanDifferenceUpper pValue
__________ _________ ______________ ___________________ ___________________ _________
"Gourmet" "Generic" 2.25 1.341 3.159 4.402e-05
"National" "Generic" 0.75 -0.15904 1.659 0.11012
Each row of m
contains a p-value for the null hypothesis that the means of the groups in columns Group1
and Group2
are not significantly different. The p-value in the first row is small enough to reject the null hypothesis that the mean popcorn yield of Gourmet
is not significantly different from that of Generic
.The p-value in the second row is too large to reject the null hypothesis that the mean popcorn yield of National
is not significantly different from that of Generic
. The value for MeanDifference
is positive in the first row; therefore, the mean popcorn yield of Gourmet
is significantly higher than that of Generic
.
Compare Group Means of Two-Way ANOVA
Load the patients data.
load patients.mat
Create a table containing variables with factor values for the smoking status and physical location of patients, and the response data for systolic blood pressure.
tbl = table(Smoker,Location,Systolic)
tbl=100×3 table
Smoker Location Systolic
______ _____________________________ ________
true {'County General Hospital' } 124
false {'VA Hospital' } 109
false {'St. Mary's Medical Center'} 125
false {'VA Hospital' } 117
false {'County General Hospital' } 122
false {'St. Mary's Medical Center'} 121
true {'VA Hospital' } 130
false {'VA Hospital' } 115
false {'St. Mary's Medical Center'} 115
false {'County General Hospital' } 118
false {'County General Hospital' } 114
false {'St. Mary's Medical Center'} 115
false {'VA Hospital' } 127
true {'VA Hospital' } 130
false {'St. Mary's Medical Center'} 114
true {'VA Hospital' } 130
⋮
Perform a two-way ANOVA to test the null hypothesis that systolic blood pressure is not significantly different between smokers and non-smokers or locations.
aov = anova(tbl,"Systolic")
aov = 2-way anova, constrained (Type III) sums of squares. Systolic ~ 1 + Smoker + Location SumOfSquares DF MeanSquares F pValue ____________ __ ___________ ______ __________ Smoker 2154.4 1 2154.4 94.462 5.9678e-16 Location 46.064 2 23.032 1.0099 0.36811 Error 2189.5 96 22.807 Total 4461.2 99 Properties, Methods
The p-values indicate that enough evidence exists to conclude that smoking status has a significant effect on blood pressure. However, not enough evidence exists to conclude that physical location has a significant effect.
Investigate the mean differences between the response data from each group.
m = multcompare(aov,["Smoker","Location"])
m=15×6 table
Group1 Group2 MeanDifference MeanDifferenceLower MeanDifferenceUpper pValue
_______________________________________ _______________________________________ ______________ ___________________ ___________________ __________
Smoker Location Smoker Location
______ _____________________________ ______ _____________________________
false {'County General Hospital' } true {'County General Hospital' } -9.935 -12.908 -6.9623 7.6385e-15
false {'County General Hospital' } false {'VA Hospital' } 1.516 -1.6761 4.708 0.73817
false {'County General Hospital' } true {'VA Hospital' } -8.419 -12.899 -3.9394 5.3456e-06
false {'County General Hospital' } false {'St. Mary's Medical Center'} 0.3721 -3.2806 4.0248 0.99968
false {'County General Hospital' } true {'St. Mary's Medical Center'} -9.5629 -14.637 -4.4886 5.0113e-06
true {'County General Hospital' } false {'VA Hospital' } 11.451 7.2101 15.692 8.3835e-11
true {'County General Hospital' } true {'VA Hospital' } 1.516 -1.6761 4.708 0.73817
true {'County General Hospital' } false {'St. Mary's Medical Center'} 10.307 5.9931 14.621 6.5271e-09
true {'County General Hospital' } true {'St. Mary's Medical Center'} 0.3721 -3.2806 4.0248 0.99968
false {'VA Hospital' } true {'VA Hospital' } -9.935 -12.908 -6.9623 7.6385e-15
false {'VA Hospital' } false {'St. Mary's Medical Center'} -1.1439 -4.8086 2.5209 0.94367
false {'VA Hospital' } true {'St. Mary's Medical Center'} -11.079 -16.058 -6.0994 6.0817e-08
true {'VA Hospital' } false {'St. Mary's Medical Center'} 8.7911 4.3482 13.234 1.5297e-06
true {'VA Hospital' } true {'St. Mary's Medical Center'} -1.1439 -4.8086 2.5209 0.94367
false {'St. Mary's Medical Center'} true {'St. Mary's Medical Center'} -9.935 -12.908 -6.9623 7.6385e-15
Each p-value corresponds to the null hypothesis that the means of groups in the same row are not significantly different. The table includes six p-values greater than 0.05, corresponding to the six pairs of groups with the same smoking status value. Therefore, systolic blood pressure is not significantly different between groups with the same smoking status value.
Input Arguments
aov
— Analysis of variance results
anova
object
Analysis of variance results, specified as an anova
object.
The properties of aov
contain the factors and response data used by
multcompare
to compute the difference in means.
factors
— Factors used to group response data
string vector | cell array of character vectors
Factors used to group the response data, specified as a string vector or cell array of
character vectors. The multcompare
function groups the response
data by the combinations of values for the factors in factors
. The
factors
argument must be one or more of the names in
aov.FactorNames
.
Example: ["g1","g2"]
Data Types: string
| cell
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Example: Alpha=0.01,CriticalValueType="dunnett",Approximate=true
sets
the significance level of the confidence intervals to 0.01 and uses an approximation of
Dunnett's critical value to calculate the p-values.
Alpha
— Significance level
0.05
(default) | scalar value in the range (0,1)
Significance level for the estimates, specified as a scalar value in the range
(0,1). The confidence level of the confidence intervals is . The default value for Alpha
is
0.05
, which returns 95% confidence intervals for the
estimates.
Example: Alpha=0.01
Data Types: single
| double
CriticalValueType
— Critical value type
"tukey-kramer"
(default) | "hsd"
| "dunn-sidak"
| "bonferroni"
| "scheffe"
| "dunnett"
| "lsd"
Critical value type used by the multcompare
function to calculate
p-values, specified as one of the options in the following table.
Each option specifies the statistical test that multcompare
uses to
calculate the critical value.
Option | Statistical Test |
---|---|
"tukey-kramer" (default) | Tukey-Kramer test |
"hsd" | Honestly Significant Difference test — Same as
"tukey-kramer" |
"dunn-sidak" | Dunn-Sidak correction |
"bonferroni" | Bonferroni correction |
"scheffe" | Scheffé test |
"dunnett" | Dunnett's test — Can be used only when aov
is a one-way anova object or when a single factor
is specified in factors . For Dunnett's test,
the control group is selected in the generated plot and cannot be
changed. |
"lsd" | Stands for Least Significant Difference and uses the critical value for a plain t-test. This option does not protect against the multiple comparisons problem unless it follows a preliminary overall test such as an F-test. |
Example: CriticalValueType="dunn-sidak"
Data Types: char
| string
Approximate
— Indicator to compute Dunnett critical value approximately
true
or 1
| false
or 0
Indicator to compute the Dunnett critical value approximately, specified as a numeric
or logical 1
(true
) or 0
(false
). You can compute the Dunnett critical value
approximately for speed. The default for Approximate
is
true
for an N-way ANOVA with N greater than two, and
false
otherwise. This argument is valid only when
CriticalValueType
is "dunnett"
.
Example: Approximate=true
Data Types: logical
ControlGroup
— Index of control group factor value
1 (default) | positive integer
Index of the control group factor value for Dunnett's test, specified as a positive
integer. Factor values are indexed by the order in which they appear in
aov.ExpandedFactorNames
. This argument is valid only when
CriticalValueType
is "dunnett"
.
Example: ControlGroup=3
Data Types: single
| double
Output Arguments
m
— Multiple comparison procedure results
table
Multiple comparison procedure results, returned as a table. The table
m
has the following variables:
Group1
— Values of the factors in the first comparison groupGroup2
— Values of the factors in the second comparison groupMeanDifference
— Difference in mean response between the observations inGroup1
and the observations inGroup2
MeanDifferenceLower
— 95% lower confidence bound on the mean differenceMeanDifferenceUpper
— 95% upper confidence bound on the mean differencepValue
— p-value indicating whether or not the mean ofGroup1
is significantly different from the mean ofGroup2
If two or more factors are provided in factors
, the columns
Group1
and Group2
contain tables of values for
the factors of the groups being compared.
References
[1] Hochberg, Y., and A. C. Tamhane. Multiple Comparison Procedures. Hoboken, NJ: John Wiley & Sons, 1987.
[2] Milliken, G. A., and D. E. Johnson. Analysis of Messy Data, Volume I: Designed Experiments. Boca Raton, FL: Chapman & Hall/CRC Press, 1992.
[3] Searle, S. R., F. M. Speed, and G. A. Milliken. “Population marginal means in the linear model: an alternative to least-squares means.” American Statistician. 1980, pp. 216–221.
[4] Dunnett, Charles W. “A Multiple Comparison Procedure for Comparing Several Treatments with a Control.” Journal of the American Statistical Association, vol. 50, no. 272, Dec. 1955, pp. 1096–121.
[5] Krishnaiah, Paruchuri R., and J. V. Armitage. "Tables for multivariate t distribution." Sankhyā: The Indian Journal of Statistics, Series B (1966): 31-56.
Version History
Introduced in R2022b
See Also
plotComparisons
| groupmeans
| anova
| One-Way ANOVA | Two-Way ANOVA | N-Way ANOVA
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