Real Statistics Functions
The Real Statistics Resource Pack provides the following array functions associated with MANOVA for the range R1 with data in standard form (without column headings).
MANOVA_T(R1) = T
MANOVA_H(R1) = H
MANOVA_E(R1) = E
In addition, the resource pack provides the following functions associated with the Wilk’s Lambda Test:
MANOVA_WilksLambda(R1) = Λ
MANOVA_Wilksdf1(R1) = df1
MANOVA_Wilksdf2(R1) = df2
MANOVA_WilksF(R1) = F
MANOVA_WilksTest(R1) = p-value
For example, MANOVA_WilksLambda(A4:D35) = 0.4894 for the data in Example 1 of Manova Basic Concepts.
It also provides the following functions associated with the Hotelling-Lawley Trace Test:
MANOVA_Hoteldf1(R1) = df1
MANOVA_Hoteldf2(R1) = df2
MANOVA_HotelF(R1) = F
MANOVA_HotelTest(R1) = p-value
It also provides the following functions associated with the Pillai-Bartlett Trace Test:
MANOVA_PillaiTrace(R1) = V
MANOVA_Pillaidf1(R1) = df1
MANOVA_Pillaidf2(R1) = df2
MANOVA_PillaiF(R1) = F
MANOVA_PillaiTest(R1) = p-value
The resource pack also supplies the following array functions each of which outputs a 5 × 1 array with the appropriate Manova statistic, df1, df2, F statistic and appropriate p-value.
Finally, the resource pack contains the following functions regarding Roy’s Largest Root:
MANOVA_RoyRoot(R1, b) = largest eigenvalue λp of HE-1 if b = TRUE (default) and = if b = FALSE
Real Statistics Data Analysis Tool
The Real Statistics Resource Pack provides the MANOVA data analysis tool. This tool can be employed for the analysis of Example 1 of Manova Basic Concepts as follows:
Step 1: Press Ctrl-m to open the dialog box for supplemental data analyses and double click on the Analysis of Variance (or Multivariate Analyses) option
Step 2: Choose the MANOVA data analysis option from the dialog box that appears and click on the OK button
Step 3: The dialog box shown in Figure 1 will now appear
Figure 1 – Single Factor Manova dialog box
Step 4: Insert the Input Range (in standard format). For Example 1 of Manova Basic Concepts, insert A3:D35 in the Input Range (using the data from Figure 1 of Manova Basic Concepts, including the column headings). Click on the Regular analysis type and all the desired options (for now we select the first three options) and click on the OK button.
Step 5: The output shown in Figure 2 appears.
Figure 2 – Manova analysis for Example 1
The output from the various tests (range F6:L9) is the same as we obtained in Figure 6, 7 and 8 of Manova Basic Concepts. The results for T, H and E (range N5:P18) are the same as we obtained in Figure 5 of Manova Basic Concepts.
The covariance matrices for each group as well as the pooled covariance matrix and total correlation matrix are shown in range R5:T31. E.g. the covariance matrix for the clay group (range R4:T6) can be computed by the supplemental formula COV(B28:D35).
Definition 1: If S1, S2, …, Sm are k × k group covariance matrices where each group g has ng elements, then the pooled covariance matrix S is
For Example 1 of Manova Basic Concepts, the pooled group matrix is shown in range R24:T26 of Figure 2. It can also be computed by the array formula R4:T6+R9:T11+R14:T16+R19:T121.
Observation: The total SSCP can be computed as the (total) covariance matrix of the sample times dfT = n – 1. Similarly the error SSCP can be computed by the pooled covariance matrix times dfE = n – m, i.e.
T = CovT ∙ (n – 1)
E = CovPooled ∙ (n – m)
In fact, multiplying range R24:T26 by n – m = 32 – 4 = 28 does indeed yield the E matrix as shown in range L14:N16 of Figure 5 of Manova Basic Concepts.
More Real Statistics Functions
The Real Statistics Resource Pack provides the following array functions where range R1 contains data in standard form (with or without column headings) and s is a string which presumably specifies a group (in the first column of R1):
ExtractRows(R1, s, b) = the array which contains all the elements in range R1 for the group labeled s. If b is set to True, the first row of R1 is included in the output (presumably column headings) even if there is no match. If b is omitted it defaults to True.
ExtractCov(R1, s) = the covariance matrix for the group labeled s, based on the data in range R1. If s is the empty string “” or is omitted then the pooled covariance matrix of R1 is returned.
Note that when calculating the pooled covariance matrix using ExtractCov(R1) where R1 contains column headings, it is important that the first cell in R1 be empty (otherwise this will be interpreted as representing another group).
The Real Statistics Resource Pack also provides the following array function where range R1 contains data in standard form (with column headings) and s is a string which specifies a column heading (i.e. an entry in the first row of R1), which selects one of the dependent variables.
ExtractCol(R1, s) = an array which consists of all the data in R1 for the dependent variable identified by s, now organized by columns with one column for each group. You should highlight a range with g columns where g = the number of groups.
Observation: We can use the ExtractCov supplemental function to calculate the group and pooled covariance matrices displayed in Figure 2. E.g. the covariance matrix for the clay group (range R4:T6) can be computed by ExtractCov(A3:D35,”clay”) or ExtractCov(A4:D35,”clay”). The pooled covariance matrix (range R24:T26) can be computed by ExtractCov(A3:D35) or ExtractCov(A4:D35).
Observation: We can use ExtractRows to extract only those data elements in a particular group. Referring to the data in Figure 1 of Manova Basic Concepts, we can extract the data for the clay group using the formula ExtractRows(A3:D35,”clay”) as shown on the left side of Figure 3. Clearly the covariance matrix for the clay group can also be calculated by COV(J4:L11).
Figure 3 – Use of ExtractRows and ExtractCol functions
We can use ExtractCol to extract only those data elements for a particular independent variable, organized by group with one group per column. Referring again to the data in Figure 1 of Manova Basic Concepts, we can extract the data for water using the formula ExtractCol(A3:D35,”water”) as shown on the right side of Figure 3.
Observation: Referring back to Figure 2, we will discuss the eta-squared effect size in Manova Effect Size and the correlation matrix in Manova Assumptions. We also describe the other MANOVA data analysis tool options (as shown in Figure 1) in Manova Follow-up Anova and Manova Follow-up Contrasts.