3.1 INTRODUCTION

This chapter generalizes the ideas developed in Chapter 2. Let there be k populations and let k-independent random samples be taken from these k populations. Let Y_iα be the p-dimensional response vector on the αth unit from the ith population, α = 1, 2, …, N_i; i = 1, 2, …, k.

Let Y_iα ~ IMN_p(μ_i, ∑ _i). Let . It is not necessary to have the observations taken from k populations. One may take N homogeneous units, randomly subdivide them into k sets of sizes N₁, N₂, …, N_k, assign the k treatments such that the ith treatment is assigned to each of the N_i experimental units of the ith set, and take p responses at the same time or at different time intervals. In either case, three kinds of null hypotheses are tested:

1. Parallelism or interaction of the populations and responses
2. Equality of population or group effects
3. Equality of response or period effects

Let μ′_i = (μ_i1, μ_i2, …, μ_ip). The three kinds of null hypotheses can be represented as follows:

(3.1.1)

(3.1.2)

(3.1.3)

To test these three null hypotheses, we initially assume that ∑ ₁ = ∑ ₂ = … = ∑ _k (=∑,  say), and this assumption will be relaxed in Section 3.6. Further, if ∑ satisfies the sphericity condition, the conclusions can be drawn from the univariate ANOVA for a mixed model. Otherwise, multivariate methods can be used to analyze the data.

Test for the equality of dispersion matrices of the k populations will be discussed in Section 3.2. The univariate analysis of this problem will be given in Section 3.3, while the multivariate methods will be presented in Section 3.4. The methods discussed in Sections 3.2–3.4 will be illustrated in Section 3.5. The results with unequal dispersion matrices will be discussed in Section 3.6. Finally, in Section 3.7, we will discuss the analysis with ordered categorical responses.

3.2 Test for the Equality of Dispersion Matrices and Sphericity Condition Of k-Dispersion Matrices

As indicated in Section 3.1, let Y_iα ~ IMN_p(μ_i, ∑_i). Let P₁ be a p–1 × p matrix such that

(3.2.1)

is an orthogonal matrix. Then,

(3.2.2)

where

(3.2.3)

The null hypothesis

(3.2.4)

is equivalent to

(3.2.5)

Let and . The likelihood function for testing the null hypothesis (3.2.5) is

(3.2.6)

with likelihood ratio criterion

(3.2.7)

where the numerator is the maximum of the likelihood function over ω : {υ_i, ψ₀|–∞ < υ_ij < ∞,  i = 1, 2, …, k,  j = 1, 2, …, p – 1;  d > 0} and the denominator is the maximum of the likelihood function over Ω: {υ_i, ψ_i|–∞ < υ_ij < ∞,  i = 1, 2, …, k,  j = 1, 2, …, p – 1;  ψ_i is positive definite}.

After some algebra, we obtain

(3.2.8)

and Equation (3.2.7) simplifies to

(3.2.9)

Following Bartlett (1937), we modify the test statistic λ to

(3.2.10)

where n_i = N_i–1 and .

Letting w(A;f, ∑) to be the Wishart density function of A with f degrees of freedom and population dispersion matrix ∑,

(3.2.11)

and by using the properties of Wishart and χ² distributions, we can show that

(3.2.12)

Using an asymptotic distribution of the likelihood ratio criterion of Box (1949), we can verify that

(3.2.13)

where

(3.2.14)

Thus, asymptotically, the critical region for testing the null hypothesis (3.2.5) is

(3.2.15)

where is the upper α percentile point of the χ² distribution with (k–1)p(p–1)/2 degrees of freedom.

3.3 Univariate ANOVA for k-Sample RMD

Let Y_iαβ be the βth response on the αth unit of the ith sample (β = 1, 2,…, p;  α = 1, 2, …, N_i;  i = 1, 2, …, k). Clearly, Y′_iα = (Y_iα1, Y_iα2, …, Y_iαp).

We assume the linear model

(3.3.1)

where μ is the general mean; γ_i is the ith population effect if the sampling is done from k populations, or it is the effect of the ith treatment used on the N_i units; s_iα is the effect of the αth unit in the ith group; τ_β is the βth response effect; δ_iβ is the interaction effect of the ith group (or treatment) with the βth response; and e_iαβ are random errors. The model (3.3.1) is a mixed effects model in which s_iα and e_iαβ are random effects. Putting e′_iα = (e_iα1, e_iα2, …, e_iαp), it is further assumed that

(3.3.2)

where ∑ satisfies the sphericity condition.

Letting the sum of squares arising from model (3.3.1) are the following:

Using Lemmas 2.3.1 and 2.3.2 and calling f as the sum of all elements of ∑, that is, f = J_1,p ∑ J_p,1, , , S₃/f ~ χ²(N - k), , and S₅/d ~ χ²((p - 1)(N - k)). Further, the pairs of quadratic forms (S₁, S₅), (S₂, S₃), and (S₄, S₅) are independently distributed.

The noncentrality parameters δ₁, δ₂, and δ₃ can be evaluated as in Section 2.3 to get

where τ ′ = (τ₁, τ₂, …, τ_p), γ ′ = (γ₁, γ₂, …, γ_k), and δ ′ = (δ₁₁, δ₁₂, …, δ_1p, …, δ_k1, δ_k2, …, δ_kp).

Thus,

(3.3.3)

where is a noncentral F distribution, which reduces to the central F distribution under the null hypothesis (3.1.1). Thus, the critical region for testing the interactions between the groups and the responses given by the null hypothesis (3.1.1) is

(3.3.4)

Similarly, we note that the critical region for testing the equality of group effects given by the null hypothesis (3.1.2) is

(3.3.5)

and the critical region for testing the equality of response effects given by the null hypothesis (3.1.3) is

(3.3.6)

The analysis resulting from the earlier tests can be summarized in the k-sample RMD ANOVA (Table 3.3.1).

One can perform standard multiple comparison tests like Scheffe’s to test individual contrasts of responses or groups. When the sphericity condition is not satisfied for the common dispersion matrix, one can conservatively test the F statistics for responses with numerator and denominator degrees of freedom 1 and (p–1)(N–k).

3.4 Multivariate Methods for k-Sample RMD

Letting Y_iα to be the p-component response vector on the αth observation (or unit) taken from the ith population (or treated with ith treatment), it is assumed that Y_iα ∼ IMN_p(μ_i, ∑) for α = 1, 2, …, N_i;  i = 1, 2, …, k. Let L be a (p–1) × p matrix as defined in (2.4.1) and let

(3.4.1)

Then, clearly

(3.4.2)

where

(3.4.3)

and

(3.4.4)

The null hypothesis (3.1.1) is equivalent to

(3.4.5)

This is the standard multivariate ANOVA problem discussed in Rao (1973), Morrison (1976), and Anderson (2003) and can be solved using test statistics involving determinants or maximum eigenvalues of functions of matrices representing the hypotheses and error. Let

(3.4.6)

and

(3.4.7)

Put

(3.4.8)

and

(3.4.9)

The matrices H and E are the hypothesis and error matrices for testing the null hypothesis (3.4.5).

Let s = min(k – 1, p – 1) and let c_s be the greatest eigenvalue of HE^-1. The upper percentage points of the greatest root distribution of the matrix H(H + E)^- 1 have been computed by Heck (1960), Pillai and Bantegu (1959), and Pillai (1964, 1965, 1967). The maximum eigenvalue c_s of HE^-1 is related to the maximum eigenvalue θ_s of H(H + E)^- 1 by the relation

(3.4.10)

Using , , the upper α percentile point c_α(s, m, n) of the distribution θ_s can be obtained from Heck’s charts or Pillai’s tables, and the critical region for testing the null hypothesis (3.4.5) is

(3.4.11)

Alternatively, one notes that

(3.4.12)

and can construct critical regions accordingly for testing the null hypothesis (3.4.5).Letting

(3.4.13)

it follows that the one-dimensional random variable U_iα has

(3.4.14)

and the null hypothesis (3.1.2) is based on the F statistic with (k – 1) and (N – k) degrees of freedom given by

(3.4.15)

where

(3.4.16)

and

(3.4.17)

This can be summarized in Table 3.4.1.

Source	d.f.	S.S.	M.S.	F
Between groups	k – 1		MSb	MSb/MSe
Within groups	N – k		MSe
Total	N – 1

The critical region for testing the null hypothesis (3.1.2) is thus

(3.4.18)

where F is the F value from the ANOVA Table 3.4.1. This is the same as the test discussed for univariate ANOVA in Section 3.3.

When the null hypothesis (3.1.1) or equivalently (3.4.5) is tenable, to test the responses, let υ be the common υ₁ = υ₂ = … = υ_k. The null hypothesis (3.1.3) is then equivalent to

(3.4.19)

Now W_iα ~ IMN_p–1(υ, ψ) and testing (3.4.19) can be done as described in Section 2.4. Letting E to be defined as in Equation (3.4.9), one notes that E ~ W_p–1(N–k, ψ).

Put

(3.4.20)

and

(3.4.21)

Under the null hypothesis (3.4.19)

(3.4.22)

is distributed as an F distribution, with degrees of freedom (p – 1) and (N – k – p + 2). Thus, the critical region for testing the null hypothesis (3.1.3) is

(3.4.23)

3.5 Numerical Example

In this section, the methods discussed in Sections 3.2–3.4 will be illustrated through an example with artificial data for the situation considered in Example 1.3.2.

We assume that the dispersion matrices for the four groups are equal as we cannot test the equality of dispersion matrices for this example because the sample sizes in each group is not 1 more than the number of variables.

The following SAS program provides the necessary output:

data a;input patnum group period1 period2 period3 @@;cards;
1 1 109 115 125 2 1 110 112 135 31 105 102 100
4 2 97 120 121 5 2 126 142 148 6 2 151 197 189
7 3 122 162 138 8 3 157 152 217 9 3 212 235 213
10 3 137 183 273 11 4 225 204 197 12 4 175 181 203
;

proc glm;class group;
model period1 period2 period3=group/nouni;
*the nouni will not provide the univariate Anova for each dependent variable;
repeated period 3 profile /printe uepsdef= HF;run;
*The period indicates the label for the repeated measurements variable, 3 is the number of repeated measurement, profile is the type of transformation specified and printe provides a test for sphericity (see the SAS manual for further details);

data b;set a;
diff1=period1-period2;diff2=period2-period3;
*This is needed to study the linear and quadratic period effects across groups;
proc glm;class group;
model diff1 diff2=group/nouni;
repeated diff 2profile;run;

***

Sphericity Tests

Mauchly's

The GLM Procedure

Repeated Measures Analysis of Variance

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no period Effect

H = Type III SSCP Matrix for period

E = Error SSCP Matrix

S=1 M=0 N=2.5

MANOVA Test Criteria and F Approximations for the Hypothesis of no period^*group Effect

H = Type III SSCP Matrix for period^*group

E = Error SSCP Matrix

S=2 M=0 N=2.5

NOTE: F Statistic for Roy's Greatest Root is an upper bound.

NOTE: F Statistic for Wilks' Lambda is exact.

The GLM Procedure

Repeated Measures Analysis of Variance

Tests of Hypotheses for Between Subjects Effects

The GLM Procedure

Repeated Measures Analysis of Variance

Univariate Tests of Hypotheses for Within Subject Effects4

Adj Pr > F

Greenhouse-Geisser Epsilon 0.6485 (a11)

Huynh-Feldt Epsilon 1.0118(a12)

***

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no diff^*group Effect

H = Type III SSCP Matrix for diff^*group

E = Error SSCP Matrix

S=1 M=0.5 N=3

The GLM Procedure

Repeated Measures Analysis of Variance

Tests of Hypotheses for Between Subjects Effects

***

In the output at (a1), we have the p-value to test the sphericity condition of the dispersion matrices (Mauchly, 1940). If this p-value is more than 0.05, the sphericity assumption is not rejected. In our example, it is 0.065 and the sphericity condition is valid and hence the univariate approach can be used to test the hypothesis. At (a5) and (a8), we have the p-values for testing the period effect and the period^*group interaction effect. In our example, these are not significant. Had the sphericity condition been rejected, p-values given at (a6) and (a9) adjusting by the method of Greenhouse and Geisser or the p-values given at (a7) and (a10) adjusting by the method of Huynh and Feldt could have been used. In our problem, these p-values are also not significant. The Greenhouse and Geisser (1959) adjustment is given at (a11) and is 0.6485. The Huynh and Feldt (1976) adjustment is given at (a12) and is 1.0118.

Note that instead of the Huynh–Feldt epsilon, the Lecoutre correction of the Huynh–Feldt epsilon is displayed as a default for SAS releases from 9.22 onwards. The Huynh–Feldt epsilon’s numerator is precisely unbiased only when there are no between-subject effects. Lecoutre (1991) made a correction to the numerator of the Huynh–Feldt epsilon. However, one can still request Huynh–Feldt epsilon by using the option UEPSDEF = HF in the REPEATED statement.

In our example, the Huynh–Feldt–Lecoutre epsilon is 0.7215. The p-values for period and period^*group adjusting by the method of Huynh–Feldt–Lecoutre are 0.1167 and 0.5126.

By using the multivariate methods, the p-value at (a2) will be used to test the no period effect hypothesis and the p-value at (a3) to test the interaction effect of period and groups. The p-value at (a4) is used to test the group effect in univariate or multivariate analyses. In our example, this p-value is significant, and hence, the ABS scores differ in the four groups.

If the periods are of equal time intervals, the linear and quadratic effects of periods can be tested by using the p-values at (a13) and (a14). The linear effect equality across groups is tested by the p-value given at (a14), and it is not significant in our example. The quadratic effect equality across groups of the periods can be tested by the p-value given at (a13) and is not significant.

For further programming details, see the SAS manual.

3.6 Multivariate Methods with Unequal Dispersion Matrices

Continuing the notation used in this chapter so far, let Y_iα be distributed IMN_p(μ_i, ∑ _i) for α = 1, 2, …, N_i;  i = 1, 2, …, k. Put

where L₁ is (k–1) × k and L₂ is (p–1) × p matrices. Further, D(…) is a block diagonal matrix with entries in the diagonal position. Note that the rows of C₁μ are independent contrasts of groups and period interaction, C₂μ are independent contrasts of groups, and C₃μ are independent contrasts of periods. Following Welch (1947, 1951) and James (1951, 1954), the three null hypotheses (3.1.1), (3.1.2), and (3.1.3) can be tested using the test statistics:

Let q = kp, m₁ = (p–1)(k–1), m₂ = k–1, m₃ = p–1,

for , where Q_i is a kp square block matrix whose (u, w) is I_p for u = w = i and 0 for u ≠ w; u, w = 1, 2, …, k. The critical regions for testing the hypotheses (3.1.1), (3.1.2), and (3.1.3) are, respectively,

(3.6.1)

For some simulation results, see Keselman, Carriere, and Lix (1993). Also, see Keselman, Algina, and Kowalchuk (2001) for a review on repeated measurements designs.

Let us continue with the ABS scores data given in Table 3.5.1. Though the dispersion matrices for the four groups are not significantly different, we will still use the test discussed in this section.

data a;input patnum group period1 period2 period3 @@;cards;
1 1 109 115 125 2 1 110 112 135 3 1 105 102 100 4 2 97 120 121
5 2 126 142 148 6 2 151 197 189 7 3 122 162 138 8 3 157 152 217
9 3 212 235 213 10 3 137 183 273 11 4 225 204 197 12 4 175 181 203
;
proc sort;by group;
proc corr cov outp=stats noprint; var period1 period2 period3;by group ;run;

data stats;set stats;
proc iml;use stats;
read all into y var{period1 period2 period3};

meany=
{108.00 109.67 120.0 124.67 153.00 152.67 157.00 183. 00 210.25 200.00 192.50 200.00};

l1={1 –1 0 0,1 0 - 1 0,1 0 0 -1};
l2={1 –1 0,1 0 –1};

j1={1 1 1}; j2={1 1 1 1};
c1= l1@l2; *l1 kronecker product l2;
c2=l1@j1;
c3=j2@l2;

/***input data***/

k=4;*number of groups;
p=3;*number of periods;

/*** end of input data ***/

q=k*p;
m1=(p-1)*(k-1);
m2=k-1;
m3=p-1;

f11= q - m1;
f12 = q - m2;
f13 = q - m3;

s1=y[1:3,1:3];
s2=y[10:12,1:3];
s3=y[19:21,1:3];
s4=y[28:30,1:3];
s=block(s1,s2,s3,s4);*S=D(s1,s2,s3,s4);

T1=(C1*meany`)`*(inv(c1*s*C1`))*(c1*meany`);T2=(C2*meany`)`*(inv(c2*s*C2`))*(c2*meany`);
T3=(C3*meany`)`*(inv(c3*s*C3`))*(c3*meany`);
zero={0 0 0 ,0 0 0 ,0 0 0};
q1=block(I(3),zero,zero,zero);*create a block matrix;
q2=block(zero,I(3),zero,zero);
q3=block(zero,zero,I(3),zero);
q4=block(zero,zero,zero,I(3));

/**A1**/

d11= (s*c1`)*(inv(c1*s*c1`))*(c1*q1);
a11=(trace(d11*d11)+(trace(d11))**2)/2;
d12= (s*c1`)*(inv(c1*s*c1`))*(c1*q2);
a12=(trace(d12*d12)+(trace(d12))**2)/2;
d13= (s*c1`)*(inv(c1*s*c1`))*(c1*q3);
a13=(trace(d13*d13)+(trace(d13))**2)/3;
d14= (s*c1`)*(inv(c1*s*c1`))*(c1*q4);
a14=(trace(d14*d14)+(trace(d14))**2)/1;a1=(a11+a12+a13+a14)/2;

/**A2**/
d21= (s*c2`)*(inv(c2*s*c2`))*(c2*q1);
a21=(trace(d21*d21)+(trace(d21))**2)/2;
d22= (s*c2`)*(inv(c2*s*c2`))*(c2*q2);
a22=(trace(d22*d22)+(trace(d22))**2)/2;
d23= (s*c2`)*(inv(c2*s*c2`))*(c2*q3);
a23=(trace(d13*d13)+(trace(d13))**2)/3;
d24= (s*c2`)*(inv(c2*s*c2`))*(c2*q4);
a24=(trace(d14*d14)+(trace(d14))**2)/1;
a2=(a21+a22+a23+a24)/2;

/**A3**/
d31= (s*c3`)*(inv(c3*s*c3`))*(c3*q1);
a31=(trace(d31*d31)+(trace(d31))**2)/2;d32= (s*c3`)*(inv(c3*s*c3`))*(c3*q2);
a32=(trace(d32*d32)+(trace(d32))**2)/2;
d33= (s*c3`)*(inv(c3*s*c3`))*(c3*q3);
a33=(trace(d33*d33)+(trace(d33))**2)/3;
d34= (s*c3`)*(inv(c3*s*c3`))*(c3*q4);
a34=(trace(d34*d34)+(trace(d34))**2)/1;
a3=(a31+a32+a33+a34)/2;

f21=((q-m1)*(q-m1+2))/3*a1;
f22=((q-m2)*(q-m2+2))/3*a2;
f23=((q-m3)*(q-m3+2))/3*a3;
e1=q-m1+2*a1-((6*a1)/(q-m1+2));
e2=q-m2+2*a2-((6*a2)/(q-m2+2));
e3=q-m3+2*a3-((6*a3)/(q-m3+2));

p1=1-probf(t1/e1,f11,f21);*p-value for the interaction;
p2=1-probf(t2/e3,f12,f22);*p-value for the groups;
p3=1-probf(t3/e3,f13,f23); *p-value for the responses;
print p1 p2 p3;quit;run;

From the output, the p-values p1, p2, and p3 are used to test the significance of the interaction, groups, and responses. Our values indicate the interaction is not significant. Hence, we can test for the groups and responses and we observe the groups p-value is significant, whereas the responses p-value is not significant.

3.7 Analysis with Ordered Categorical Response

In some cases, the response Y_iαβ taken at βth period on the αth unit of the ith group takes any of c-ordered categorical responses for β = 1, 2, …, p; α = 1, 2 …, N_i, i = 1, 2, …, k. To illustrate, let us consider the data given in Table 3.5.1 and let us convert it into an ordered categorical data with three categories defined by a: <110; b: [110, 150]; c: ≥150 to obtain Table 3.7.1.

Let be the responses in the p periods, where = 1, 2, …, c for β = 1, 2, …, p and let be the probability of getting responses , for the ith group. For each i = 1, 2, …, k, we get c^p joint probabilities. We define the marginal probabilities:

We estimate these probabilities from the data. From Table 3.7.1, we construct Table 3.7.2, giving the marginal probabilities in parentheses.

We will now discuss the cumulative logit analysis of the marginal probabilities here, and for other models of analyses, the interested reader is referred to Kenward and Jones (1992).

We consider the (c – 1) logits

and may model

where is the general mean, is the ith group effect, is the βth period effect, and is the interaction effect of ith group and jth period effects.

However, it is preferable to assume , , and , and this assumption is known as proportional odds model. Under the proportional odds model,

For the data of Table 3.7.2, the following program produces the necessary output to draw inferences on the interaction, groups, and periods effects:

data a;input period group score freq @@;cards;
1 1 1 2 1 1 2 1 1 1 3 0 2 1 1 1 2 1 2 2 2 1 3 0 3 1 1 1 3 1 2 2 3 1 3 0 1 2 1 1 1 2 2 1 1 2 3 1

2 2 1 0 2 2 2 2 2 2 3 1 3 2 1 0 3 2 2 2 3 2 3 1 1 3 1 0 1 3 2 2 1 3 3 2 2 3 1 0 2 3 2 0 2 3 3 4

3 3 1 0 3 3 2 1 3 3 3 3 1 4 1 0 1 4 2 0 1 4 3 2 2 4 1 0 2 4 2 0 2 4 3 2 3 4 1 0 3 4 2 0 3 4 3 2

;

data b;set a;
proc logistic descending; freq freq;
model score=period group period*group; run;***

Model Convergence Status

Convergence criterion (GCONV=1E-8) satisfied.

Score Test for the Proportional Odds Assumption

***

Testing Global Null Hypothesis: BETA=0

Analysis of Maximum Likelihood Estimates

Association of Predicted Probabilities and Observed Responses

From the p-value of 0.7235 given at (b1), we observe that the proportional odds model is valid in this setting. From the value 0.897 given at (b10) for c, we observe the model is a good fit. The overall significance of the parameters is tested by the p-values given at (b2), (b3), or (b4), and all of these p-values indicate strong significance for all of the parameters. The p-value for interaction given at (b9) is not significant. The p-value for the period effects given at (b7) is not significant. However, the p-value for the groups given at (b8) indicates a borderline nonsignificance. By using the DESCENDING option in PROC LOGISTIC, the ith intercept corresponds to the ith or higher category of the response versus the categories less than i, and the p-value corresponding to the intercept i can be used to test its significance. In our problem, the p-value corresponding to (b5) is significant, indicating that the intercept term corresponding to the third category versus the first and second categories is significant. Similarly, the p-value corresponding to (b6) is not significant, indicating that the intercept corresponding to the second and third categories versus the first category is not significant.

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