3.6. Simultaneous Confidence Intervals

If H₀: LB = 0 is rejected, it may be of interest to provide the confidence intervals for the individual components of LB (or B if L= I_k+1) or the linear functions of these components. Under the assumption of the full rank of X, a set of simultaneous confidence intervals for the linear combinations of the type c′LBd corresponding to the linear hypothesis H₀: LB = 0 can be constructed.

Noting that H₀^(c,d) : (c′LBd=0 is true for all c and d if and only if H₀ is true, we can write H₀ as the intersection of all such H₀^(c,d) : (c′LBd=0. Testing of H₀^(c,d) : c′LBd=0 can be done using the appropriate F test. Let the corresponding F statistic be F^(c,d) and let F_α be the cutoff point. Then, H₀ is not rejected if and only if all H₀^(c,d) are not rejected, that is, if and only if max_c,d F^(c,d) ≤ F_α. In fact, the max_c,d F^(c,d) is equal to (n−k−1)λ_max, where λ_max is the Roy's largest root test statistic, defined earlier, corresponding to H₀ indicated above. Thus 100(1−α)% simultaneous confidence intervals for all linear combinations c′LBd are given by

where E is the error sums of squares and product matrix and λ_α, the cutoff point for λ_max, is such that P[λ_max≤λ_α] = 1−α.

The tables for the cutoff points λ_α are available in Pillai (1960). Alternatively, the F statistics approximation from λ_max which is reported in Table 3.2 can be used. For Program 3.5, we follow the latter alternative. Thus the (1−α)100% cutoff point λ_α can be approximated by

The calculations of the confidence intervals for a choice of c and d are illustrated in the following example using the IML procedure (see Program 3.5). Note that the matrices , (X′X)⁻¹ and E are available as the outputs of PROC REG or PROC GLM. Selected parts of the output of Program 3.5 are shown in Output 3.5.

EXAMPLE 4

Confidence Intervals for Cork Deposits in Cork Data Chapter 1 states that our interest in these data is in discovering if cork deposits are uniform in all four directions. Also, recall that an appropriate set of transformations of variables to do this would be

z₁ = y₁−y₂ + y₃ − y₄, z₂=y₃ − y₄, z₃=y₁−y₃.

To construct the (1−α)100% simultaneous confidence intervals on the corresponding means μ₁−μ₂+μ₃−μ₄, μ₃−μ₄ and μ₁−μ₃, we write each of these means as c′LBd with the respective choices of d as the columns of

and with c=1, L=1, and B=(μ₁ μ₂ μ₃ μ₄).

In Program 3.5 the matrices c, d, L and matrices , X′X (both obtained from the output of PROC REG) and the matrix E (obtained from the output of PROC GLM) are explicitly specified.

In the present context,

r_t=Rank(H+E) = p = 4,

r=Rank (L)=Rank([1])=1,

n=28,

k=0,

s=min(r,r_t)=1,

h=max(r,r_t)=4,

m₁=(|r−r_t|−1)/2=1 and

m₂=(n−k−r_t−2)/2=11.

As a result follows an exact F(4,24) distribution (since s=1) F(h,n−k−h+r−1). If α=0.05, then using Equation 3.17 we can compute the 95% cutoff point λ_0.05 as F_0.05(4,24)=0.4627.

/* Program 3.5 */

options ls=64 ps=45 nodate nonumber;
    title1 'Output 3.5';
    proc iml;
    alpha = .05 ;
    n = 28;
    /*
    Calculations for simultaneous confidence intervals are shown below.
       r_t =Rank (H+E)=p=4, Rank(L)=1, k=0
       df of error matrix = dferror = 27
       s = min(r, r_t) = 1, h=max(r,r_t)=4
       m1 = .5(|r−r_t| − 1) = 1
       m2 = .5(n−k−r_t−2) = 11.
       lambda  = [h/(n−k−h+r−1)]F(alpha,h, n-k-h+r−1)
    */
    r_t=4;
    r=1;
    k=0;
    s = min(r, r_t);
    h=max(r,r_t);
    m1 = .5*(abs(r-r_t) − 1);
    m2 = .5*(n−k−r_t−2);
    lambda  = (h/(n−k−h+r−1))*finv(1−alpha,h,n-k-h+r−1);
    cutoff = sqrt(lambda);
    /*
    Cut-off point for Bonferroni intervals will be computed as follows:
      dferror = 27 ; * dferror=n−1;
      g=3.0; * g is the no. of comparisons;
      cutoff = tinv(1−(alpha/(2*g)),dferror);
      cutoff=cutoff/sqrt(dferror);
    */
    xpx = {28};
    e = {7840.9643 6041.3214 7787.8214 6109.3214,
         6041.3214 5938.1071 6184.6071 4627.1071,
         7787.8214 6184.6071 9450.1071 7007.6071,
         6109.3214 4627.1071 7007.6071 6102.1071};
    bhat = {50.535714 46.178571 49.678571 45.178571};
    l = {1} ;
    c={1};
    d1={1,−1,1,−1};
    d2={0,0,1,−1};
    d3={1,0,−1,0};
    clbhatd1=c`*l*bhat*d1;
    clbhatd2=c`*l*bhat*d2;
    clbhatd3=c`*l*bhat*d3;
    cwidth1=cutoff*sqrt((c`*l*(inv(xpx))*l`*c)*(d1`*e*d1));

cwidth2=cutoff*sqrt((c`*l*(inv(xpx))*l`*c)*(d2`*e*d2));
    cwidth3=cutoff*sqrt((c`*l*(inv(xpx))*l`*c)*(d3`*e*d3));
    cl11=clbhatd1−cwidth1;
    cl12=clbhatd1+cwidth1;
    cl21=clbhatd2−cwidth2;
    cl22=clbhatd2+cwidth2;
    cl31=clbhatd3-cwidth3;
    cl32=clbhatd3+cwidth3;
    print 'Simultaneous Confidence Intervals';
    print 'For first contrast: (' cl11', '  cl12 ')';
    print 'For second contrast:(' cl21', ' cl22 ')';
    print 'For third contrast: (' cl31', ' cl32 ')';
    run;

In the IML procedure code given in Program 3.5 it is necessary only to specify r_t, r, n, k and α explicitly. The rest of the parameters and the (approximate) value of λ_α are computed by the program.

Example 3.5. Output 3.5

Simultaneous Confidence Intervals


                                    CL11         CL12
         For first contrast: ( 1.2786666,  16.435619 )


                                    CL21         CL22
         For second contrast:( −0.539816,  9.5398157 )


                                    CL31         CL32
         For third contrast: ( −4.467176,  6.1814617 )

Output 3.5 shows that three confidence intervals are (1.2787, 16.4356), (−0.5398, 9.5398) and (−4.4672, 6.1815) respectively. It may be noted that the first of these intervals does not contain zero and therefore, it is this contrast, namely the difference between the average (or sums of) deposits in (N,S) and (E,W) direction, which caused the rejection of the hypothesis of uniform cork deposits in all four directions.

Hotelling's T² offers another choice of simultaneous confidence intervals. See Johnson and Wichern (1998, p. 239) for further details. These simultaneous confidence intervals have the drawback of being too wide. However, if the interest is in only a few specific linear combinations, it is possible to provide corresponding confidence intervals, which are shorter, using Bonferroni's inequality. These confidence intervals are based on the usual Student's t test for the associated univariate linear hypothesis. For instance, if c_i′LBd_i, i = 1, ..., g are the g linear functions of interest, then 100(1−α)% Bonferroni's intervals are

where α₁ +... +α_g = α and t_ν (δ) is the 100(1−δ)% upper cutoff point from a t-distribution with ν degrees of freedom. Note that in order to compute these intervals the IML procedure as shown in Program 3.5 can be used with λ_d^1/2 replaced by t_n−k−1(α_i/2), for i = 1, ..., g respectively. The corresponding statements have been included in Program 3.5 with α_i=α/3 but have been commented out. To obtain the Bonferroni intervals, replace these statements with those corresponding to simultaneous confidence intervals.