anovan
N-way analysis of variance
Syntax
Description
returns
a vector of p-values for multiway (n-way)
ANOVA using additional options specified by one or more p
= anovan(y
,group
,Name,Value
)Name,Value
pair
arguments.
For example, you can specify which predictor variable is continuous, if any, or the type of sum of squares to use.
[
returns a p
,tbl
,stats
]
= anovan(___)stats
structure
that you can use to perform a multiple comparison test, which
enables you to determine which pairs of group means are significantly
different. You can perform such a test using the multcompare
function by providing the stats
structure
as input.
Examples
Three-Way ANOVA
Load the sample data.
y = [52.7 57.5 45.9 44.5 53.0 57.0 45.9 44.0]'; g1 = [1 2 1 2 1 2 1 2]; g2 = {'hi';'hi';'lo';'lo';'hi';'hi';'lo';'lo'}; g3 = {'may';'may';'may';'may';'june';'june';'june';'june'};
y
is the response vector and g1
, g2
, and g3
are the grouping variables (factors). Each factor has two levels ,and every observation in y
is identified by a combination of factor levels. For example, observation y(1)
is associated with level 1 of factor g1
, level 'hi'
of factor g2
, and level 'may'
of factor g3
. Similarly, observation y(6)
is associated with level 2 of factor g1
, level 'hi'
of factor g2
, and level 'june'
of factor g3
.
Test if the response is the same for all factor levels.
p = anovan(y,{g1,g2,g3})
p = 3×1
0.4174
0.0028
0.9140
In the ANOVA table, X1
, X2
, and X3
correspond to the factors g1
, g2
, and g3
, respectively. The p-value 0.4174 indicates that the mean responses for levels 1 and 2 of the factor g1
are not significantly different. Similarly, the p-value 0.914 indicates that the mean responses for levels 'may'
and 'june'
, of the factor g3
are not significantly different. However, the p-value 0.0028 is small enough to conclude that the mean responses are significantly different for the two levels, 'hi'
and 'lo'
of the factor g2
. By default, anovan
computes p-values just for the three main effects.
Test the two-factor interactions. This time specify the variable names.
p = anovan(y,{g1 g2 g3},'model','interaction','varnames',{'g1','g2','g3'})
p = 6×1
0.0347
0.0048
0.2578
0.0158
0.1444
0.5000
The interaction terms are represented by g1*g2
, g1*g3
, and g2*g3
in the ANOVA table. The first three entries of p
are the p-values for the main effects. The last three entries are the p-values for the two-way interactions. The p-value of 0.0158 indicates that the interaction between g1
and g2
is significant. The p-values of 0.1444 and 0.5 indicate that the corresponding interactions are not significant.
Two-Way ANOVA for Unbalanced Design
Load the sample data.
load carbig
The data has measurements on 406 cars. The variable org
shows where the cars were made and when
shows when in the year the cars were manufactured.
Study how the mileage depends on when and where the cars were made. Also include the two-way interactions in the model.
p = anovan(MPG,{org when},'model',2,'varnames',{'origin','mfg date'})
p = 3×1
0.0000
0.0000
0.3059
The 'model',2
name-value pair argument represents the two-way interactions. The p-value for the interaction term, 0.3059, is not small, indicating little evidence that the effect of the time of manufacture (mfg date
) depends on where the car was made (origin
). The main effects of origin and manufacturing date, however, are significant, both p-values are 0.
Multiple Comparisons for Three-Way ANOVA
Load the sample data.
y = [52.7 57.5 45.9 44.5 53.0 57.0 45.9 44.0]'; g1 = [1 2 1 2 1 2 1 2]; g2 = ["hi" "hi" "lo" "lo" "hi" "hi" "lo" "lo"]; g3 = ["may" "may" "may" "may" "june" "june" "june" "june"];
y
is the response vector and g1
, g2
, and g3
are the grouping variables (factors). Each factor has two levels, and every observation in y
is identified by a combination of factor levels. For example, observation y(1)
is associated with level 1 of factor g1
, level hi
of factor g2
, and level may
of factor g3
. Similarly, observation y(6)
is associated with level 2 of factor g1
, level hi
of factor g2
, and level june
of factor g3
.
Test if the response is the same for all factor levels. Also compute the statistics required for multiple comparison tests.
[~,~,stats] = anovan(y,{g1 g2 g3},"Model","interaction", ... "Varnames",["g1","g2","g3"]);
The p-value of 0.2578 indicates that the mean responses for levels may
and june
of factor g3
are not significantly different. The p-value of 0.0347 indicates that the mean responses for levels 1
and 2
of factor g1
are significantly different. Similarly, the p-value of 0.0048 indicates that the mean responses for levels hi
and lo
of factor g2
are significantly different.
Perform a multiple comparison test to find out which groups of factors g1
and g2
are significantly different.
[results,~,~,gnames] = multcompare(stats,"Dimension",[1 2]);
You can test the other groups by clicking on the corresponding comparison interval for the group. The bar you click on turns to blue. The bars for the groups that are significantly different are red. The bars for the groups that are not significantly different are gray. For example, if you click on the comparison interval for the combination of level 1
of g1
and level lo
of g2
, the comparison interval for the combination of level 2
of g1
and level lo
of g2
overlaps, and is therefore gray. Conversely, the other comparison intervals are red, indicating significant difference.
Display the multiple comparison results and the corresponding group names in a table.
tbl = array2table(results,"VariableNames", ... ["Group A","Group B","Lower Limit","A-B","Upper Limit","P-value"]); tbl.("Group A")=gnames(tbl.("Group A")); tbl.("Group B")=gnames(tbl.("Group B"))
tbl=6×6 table
Group A Group B Lower Limit A-B Upper Limit P-value
______________ ______________ ___________ _____ ___________ _________
{'g1=1,g2=hi'} {'g1=2,g2=hi'} -6.8604 -4.4 -1.9396 0.027249
{'g1=1,g2=hi'} {'g1=1,g2=lo'} 4.4896 6.95 9.4104 0.016983
{'g1=1,g2=hi'} {'g1=2,g2=lo'} 6.1396 8.6 11.06 0.013586
{'g1=2,g2=hi'} {'g1=1,g2=lo'} 8.8896 11.35 13.81 0.010114
{'g1=2,g2=hi'} {'g1=2,g2=lo'} 10.54 13 15.46 0.0087375
{'g1=1,g2=lo'} {'g1=2,g2=lo'} -0.8104 1.65 4.1104 0.07375
The multcompare
function compares the combinations of groups (levels) of the two grouping variables, g1
and g2
. For example, the first row of the matrix shows that the combination of level 1
of g1
and level hi
of g2
has the same mean response values as the combination of level 2
of g1
and level hi
of g2
. The p-value corresponding to this test is 0.0272
, which indicates that the mean responses are significantly different. You can also see this result in the figure. The blue bar shows the comparison interval for the mean response for the combination of level 1
of g1
and level hi
of g2
. The red bars are the comparison intervals for the mean response for other group combinations. None of the red bars overlap with the blue bar, which means the mean response for the combination of level 1
of g1
and level hi
of g2
is significantly different from the mean response for other group combinations.
Input Arguments
y
— Sample data
numeric vector
Sample data, specified as a numeric vector.
Data Types: single
| double
group
— Grouping variables
cell array
Grouping variables, i.e. the factors and factor levels of the observations in
y
, specified as a cell array. Each of the cells in
group
contains a list of factor levels identifying the
observations in y
with respect to one of the factors. The list within
each cell can be a categorical array, numeric vector, character matrix, string array, or
single-column cell array of character vectors, and must have the same number of elements
as y
.
$$\begin{array}{ccccccccccc}y& =& [& {y}_{1},& {y}_{2},& {y}_{3},& {y}_{4},& {y}_{5},& \cdots ,& {y}_{N}& {]}^{\prime}\\ & & & \uparrow & \uparrow & \uparrow & \uparrow & \uparrow & & \uparrow & \\ g1& =& \{& \text{'}A\text{'},& \text{'}A\text{'},& \text{'}C\text{'},& \text{'}B\text{'},& \text{'}B\text{'},& \cdots ,& \text{'}D\text{'}& \}\\ g2& =& [& 1& 2& 1& 3& 1& \cdots ,& 2& ]\\ g3& =& \{& \text{'}\text{hi}\text{'},& \text{'}\text{mid}\text{'},& \text{'}\text{low}\text{'},& \text{'}\text{mid}\text{'},& \text{'}\text{hi}\text{'},& \cdots ,& \text{'}\text{low}\text{'}& \}\end{array}$$
By default, anovan
treats all grouping
variables as fixed effects.
For example, in a study you want to investigate the effects of gender, school, and the education method on the academic success of elementary school students, then you can specify the grouping variables as follows.
Example: {'Gender','School','Method'}
Data Types: 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.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: 'alpha',0.01,'model','interaction','sstype',2
specifies anovan
to
compute the 99% confidence bounds and p-values for the main effects
and two-way interactions using type II sum of squares.
alpha
— Significance level
0.05 (default) | scalar value in the range 0 to 1
Significance level for confidence bounds, specified as the comma-separated pair
consisting of'alpha'
and a scalar value in the range 0 to 1. For a
value α, the confidence level is 100*(1–α)%.
Example: 'alpha',0.01
corresponds to 99% confidence
intervals
Data Types: single
| double
continuous
— Indicator for continuous predictors
vector of indices
Indicator for continuous predictors, representing which grouping variables should be treated
as continuous predictors rather than as categorical predictors, specified as the
comma-separated pair consisting of 'continuous'
and a vector of
indices.
For example, if there are three grouping variables and second one is continuous, then you can specify as follows.
Example: 'continuous',[2]
Data Types: single
| double
display
— Indicator to display ANOVA table
'on'
(default) | 'off'
Indicator to display ANOVA table, specified as the comma-separated pair consisting of
'display'
and 'on'
or
'off'
. When 'display'
is
'off'
, anovan
only returns the output
arguments, and does not display the standard ANOVA table as a figure.
Example: 'display','off'
model
— Type of the model
'linear'
(default) | 'interaction'
| 'full'
| integer value | terms matrix
Type of the model, specified as the comma-separated pair consisting
of 'model'
and one of the following:
'linear'
— The default'linear'
model computes only the p-values for the null hypotheses on the N main effects.'interaction'
— The'interaction'
model computes the p-values for null hypotheses on the N main effects and the $$\left(\begin{array}{c}N\\ 2\end{array}\right)$$ two-factor interactions.'full'
— The'full'
model computes the p-values for null hypotheses on the N main effects and interactions at all levels.An integer — For an integer value of k, (k ≤ N) for model type,
anovan
computes all interaction levels through the kth level. For example, the value 3 means main effects plus two- and three-factor interactions. The values k = 1 and k = 2 are equivalent to the'linear'
and'interaction'
specifications, respectively. The value k = N is equivalent to the'full'
specification.Terms matrix — A matrix of term definitions having the same form as the input to the
x2fx
function. All entries must be0
or1
(no higher powers).For more precise control over the main and interaction terms that
anovan
computes, you can specify a matrix containing one row for each main or interaction term to include in the ANOVA model. Each row defines one term using a vector of N zeros and ones. The table below illustrates the coding for a 3-factor ANOVA for factors A, B, and C.Matrix Row ANOVA Term [1 0 0]
Main term A
[0 1 0]
Main term B
[0 0 1]
Main term C
[1 1 0]
Interaction term AB
[1 0 1]
Interaction term AC
[0 1 1]
Interaction term BC
[1 1 1]
Interaction term ABC
For example, if there are three factors A, B, and C, and
'model',[0 1 0;0 0 1;0 1 1]
, thenanovan
tests for the main effects B and C, and the interaction effect BC, respectively.A simple way to generate the terms matrix is to modify the
terms
output, which codes the terms in the current model using the format described above. Ifanovan
returns[0 1 0;0 0 1;0 1 1]
forterms
, for example, and there is no significant interaction BC, then you can recompute ANOVA on just the main effects B and C by specifying[0 1 0;0 0 1]
formodel
.
Example: 'model',[0 1 0;0 0 1;0 1 1]
Example: 'model','interaction'
Data Types: char
| string
| single
| double
nested
— Nesting relationships
matrix of 0’s and 1’s
Nesting relationships among the grouping variables, specified
as the comma-separated pair consisting of 'nested'
and
a matrix M of 0’s and 1’s, i.e.M(i,j)
= 1 if variable i is nested in variable j.
You cannot specify nesting in a continuous variable.
For example, if there are two grouping variables District and School, where School is nested in District, then you can express this relationship as follows.
Example: 'nested',[0 0;1 0]
Data Types: single
| double
random
— Indicator for random variables
vector of indices
Indicator for random variables, representing which grouping
variables are random, specified as the comma-separated pair consisting
of 'random'
and a vector of indices. By default, anovan
treats
all grouping variables as fixed.
anovan
treats an interaction term as random
if any of the variables in the interaction term is random.
Example: 'random',[3]
Data Types: single
| double
sstype
— Type of sum of squares
3 (default) | 1 | 2 | 'h'
Type of sum squares, specified as the comma-separated pair consisting
of 'sstype'
and the following:
1 — Type I sum of squares. The reduction in residual sum of squares obtained by adding that term to a fit that already includes the terms listed before it.
2 — Type II sum of squares. The reduction in residual sum of squares obtained by adding that term to a model consisting of all other terms that do not contain the term in question.
3 — Type III sum of squares. The reduction in residual sum of squares obtained by adding that term to a model containing all other terms, but with their effects constrained to obey the usual “sigma restrictions” that make models estimable.
'h'
— Hierarchical model. Similar to type 2, but with continuous as well as categorical factors used to determine the hierarchy of terms.
The sum of squares for any term is determined by comparing two models. For a model containing
main effects but no interactions, the value of sstype
influences the
computations on unbalanced data only.
Suppose you are fitting a model with two factors and their interaction, and the terms appear in the order A, B, AB. Let R(·) represent the residual sum of squares for the model. So, R(A, B, AB) is the residual sum of squares fitting the whole model, R(A) is the residual sum of squares fitting the main effect of A only, and R(1) is the residual sum of squares fitting the mean only. The three sum of squares types are as follows:
Term | Type 1 Sum of Squares | Type 2 Sum of Squares | Type 3 Sum of Squares |
---|---|---|---|
A | R(1) – R(A) | R(B) – R(A, B) | R(B, AB) – R(A, B, AB) |
B | R(A) – R(A, B) | R(A) – R(A, B) | R(A, AB) – R(A, B, AB) |
AB | R(A, B) – R(A, B, AB) | R(A, B) – R(A, B, AB) | R(A, B) – R(A, B, AB) |
The models for Type 3 sum of squares have sigma restrictions imposed. This means, for example, that in fitting R(B, AB), the array of AB effects is constrained to sum to 0 over A for each value of B, and over B for each value of A.
Example: 'sstype','h'
Data Types: single
| double
| char
| string
varnames
— Names of grouping variables
X1,X2,...,XN
(default) | character matrix | string array | cell array of character vectors
Names of grouping variables, specified as the comma-separating pair consisting of
'varnames'
and a character matrix, a string array, or a cell
array of character vectors.
Example: 'varnames',{'Gender','City'}
Data Types: char
| string
| cell
Output Arguments
p
— p-values
vector
p-values, returned as a vector.
Output vector p
contains p-values for the null
hypotheses on the N main effects and any interaction terms specified.
Element p(1)
contains the p-value for the null
hypotheses that samples at all levels of factor A are drawn from the
same population; element p(2)
contains the p-value
for the null hypotheses that samples at all levels of factor B are
drawn from the same population; and so on.
For example, if there are three factors A, B,
and C, and 'model',[0 1 0;0 0 1;0 1 1]
,
then the output vector p
contains the p-values
for the null hypotheses on the main effects B and C and
the interaction effect BC, respectively.
A sufficiently small p-value corresponding to a factor suggests that at least one group mean is significantly different from the other group means; that is, there is a main effect due to that factor. It is common to declare a result significant if the p-value is less than 0.05 or 0.01.
tbl
— ANOVA table
cell array
ANOVA table, returned as a cell array. The ANOVA table has seven columns:
Column name | Definition |
---|---|
source | Source of the variability. |
SS | Sum of squares due to each source. |
df | Degrees of freedom associated with each source. |
MS | Mean squares for each source, which is the ratio
SS/df . |
Singular? | Indication of whether the term is singular. |
F | F-statistic, which is the ratio of the mean squares. |
Prob>F | The p-values, which is the probability that the
F-statistic can take a value larger than a computed
test-statistic value. anovan derives these probabilities from
the cdf of F-distribution. |
The ANOVA table also contains the following columns if at least one of the grouping
variables is specified as random using the name-value pair argument
random
:
Column name | Definition |
---|---|
Type | Type of each source; 'fixed' for a fixed effect or
'random' for a random effect. |
Expected MS | Text representation of the expected value for the mean square.
Q(source) represents a quadratic function of
source and V(source) represents the
variance of source . |
MS denom | Denominator of the F-statistic. |
d.f. denom | Degrees of freedom for the denominator of the F-statistic. |
Denom. defn. | Text representation of the denominator of the
F-statistic. MS(source) represents the
mean square of source . |
Var. est. | Variance component estimate. |
Var. lower bnd | Lower bound of the 95% confidence interval for the variance component estimate. |
Var. upper bnd | Upper bound of the 95% confidence interval for the variance component estimate. |
stats
— Statistics
structure
Statistics to use in a multiple comparison test using
the multcompare
function, returned
as a structure.
anovan
evaluates the hypothesis that the
different groups (levels) of a factor (or more generally, a term)
have the same effect, against the alternative that they do not all
have the same effect. Sometimes it is preferable to perform a test
to determine which pairs of levels are significantly different, and
which are not. Use the multcompare
function
to perform such tests by supplying the stats
structure
as input.
The stats
structure contains the fields listed
below, in addition to a number of other fields required for doing
multiple comparisons using the multcompare
function:
Field | Description |
---|---|
| Estimated coefficients |
| Name of term for each coefficient |
| Matrix of grouping variable values for each term |
| Residuals from the fitted model |
The stats
structure also contains the following fields if at least one of
the grouping variables is specified as random using the name-value pair argument
random
:
Field | Description |
---|---|
| Expected mean squares |
| Denominator definition |
| Names of random terms |
| Variance component estimates (one per random term) |
| Confidence intervals for variance components |
terms
— Main and interaction terms
matrix
Main and interaction terms, returned as a matrix. The terms
are encoded in the output matrix terms
using
the same format described above for input model
.
When you specify model
itself in this format,
the matrix returned in terms
is identical.
References
[1] Dunn, O.J., and V.A. Clark. Applied Statistics: Analysis of Variance and Regression. New York: Wiley, 1974.
[2] Goodnight, J.H., and F.M. Speed. Computing Expected Mean Squares. Cary, NC: SAS Institute, 1978.
[3] Seber, G. A. F., and A. J. Lee. Linear Regression Analysis. 2nd ed. Hoboken, NJ: Wiley-Interscience, 2003.
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