ancovar¶

Analyzes a one-way classification model with covariates.

Synopsis¶

ancovar (ni, y, x)

Required Arguments¶

int ni[] (Input): Array of length ngroup containing the number of responses for each group.
float y[] (Input): Array of length n containing the data for the response variable where n = ni[0] + ni[1] +…+ ni[ngroup-1].
float x[[]] (Input): Array of size n by ncov containing the data for the covariates.

Element	Anova Table Value
0	Degrees of freedom for model (groups + covariates).
1	Degrees of freedom for error.
2	Total (corrected) degrees of freedom.
3	Sum of squares for model (groups and covariates combined).
4	Sum of squares for error.
5	Total (corrected) sum of squares.
6	Model mean square (groups and covariates combined).
7	Error mean square.
8	F-statistic.
9	p-value.
10	\(R^2\) (in percent).
11	Adjusted \(R^2\) (in percent).
12	Estimate of the standard deviation.
13	Overall response mean.
14	Coefficient of variation (in percent).

An array of length 15 containing the one-way analysis of covariance assuming parallelism, organized as follows:

Return Value¶

Note that the p‑value is returned as 0.0 when the value is so small that all significant digits have been lost.

Optional Arguments¶

nMissing (Output)

The number of cases with missing values in x or y is returned in nMissing. Cases with any missing values are not used in the analysis.

adjAnova (Output)

An array of length 8 containing the partial sum of squares for the one-way analysis of covariance organized as follows:

`i`	`adjAnova[i]`
0	Degrees of freedom for groups after covariates.
1	Degrees of freedom for covariates after groups.
2	Sum of squares for groups after covariates.
3	Sum of squares for model (groups and covariates combined).
4	F -statistic for groups.
5	F -statistic for covariates.
6	p-value for groups.
7	p-value for covariates.

Note that the p-values are returned as 0.0 when the values are so small that all significant digits have been lost.

parallelTests (Output)

An array of length 10 containing the parallelism tests for the one-way analysis of covariance organized as follows:

`i`	parallelTests[i]
0	Extra degrees of freedom for model not assuming parallelism.
1	Degrees of freedom for error for model not assuming parallelism.
2	Degrees of freedom for error for model assuming parallelism.
3	Extra sum of squares for model not assuming parallelism.
4	Sum of squares for error for model not assuming parallelism.
5	Sum of squares for error for model assuming parallelism.
6	Mean square for `parallelTests[`0`]`.
7	Mean square for `parallelTests[`1`]`.
8	F –statistic.
9	p-value.

xymean (Output): An array of size ngroup+1 by ncov+3 containing the unadjusted means for the covariates and the response variate and the means for the response variate adjusted for the covariates. Each row for i = 0, 1, …, ngroup-1 corresponds to a group. Row ngroup contains overall statistics. The means are organized in xymean columns as follows:

Column	Description
0	Number of non-missing cases
1 thru `ncov`	Covariate means.
`ncov` + 1	Response mean.
`ncov` + 2	Response mean adjusted assuming parallelism.

coef (Output): An array of size ngroup + ncov by 4 containing statistics for the regression coefficients for the model assuming parallelism. Each row corresponds to a coefficient in the model. For i = 0, 1, …, ngroup-1, row i is for the y intercept for the i-th group. The remaining ncov rows are for the covariate coefficients. The statistics in the columns are organized as follows:

Column	Description
0	Coefficient estimate.
1	Estimated standard error of the estimate.
2	t-statistic.
3	p-value.

coefTables (Output)

An array of size ngroup by ncov+1 by 4 containing statistics for a linear regression model fitted separately for each of the ngroup treatment groups. This array can be viewed as a 3 dimensional array with ngroup rows, ncov+1 columns, and depth of 4. Each row corresponds to one of the ngroup treatment groups. Each column corresponds to the model coefficients.

For column = 0, the statistics relate to the intercept in the regression model. For column = 1, 2, …, ncov, the statistics relate to the slopes for the covariates. The depth dimension corresponds to the columns described for coef as follows:

Column	Description
0	Coefficient estimate.
1	Estimated standard error of the estimate.
2	t-statistic.
3	p-value.

regAnova (Output)

An array of size ngroup by 15 containing the analysis of variance tables for each linear regression model fitted separately to each treatment group. The 15 values in the i-th row are for treatment group i organized as follows:

`j`	*regAnova[i15+j]**
0	Degrees of freedom for regression model (covariates).
1	Degrees of freedom for error.
2	Total (corrected) degrees of freedom.
3	Sum of squares for regression model.
4	Sum of squares for error.
5	Total (corrected) sum of squares.
6	Model mean square.
7	Error mean square.
8	F-statistic.
9	p-value.
10	\(R^2\) (in percent).
11	Adjusted \(R^2\) (in percent).
12	Error standard deviation.
13	Overall response mean.
14	Coefficient of variation (in percent).

Note that the p‑value is returned as 0.0 when the value is so small that all significant digits have been lost.

rMatrix (Output)

An array of size ngroup+ncov by ngroup + ncov containing the R matrix from the QR decomposition. The R matrix is from the regression assuming parallelism.

covMeans (Output)

An array of size ngroup by ngroup containing the estimated matrix of variances and covariances for the adjusted means assuming parallelism.

covCoef (Output)

An array of size ngroup + ncov by ngroup+ncov containing the estimated matrix of variances and covariances for the coefficients in coef returned using coef.

Description¶

Function ancovar performs analyses for models that combine the features of a one-way analysis of variance model with that of a multiple linear regression model. The basic one-way analysis of covariance model is

\[y_{ij} = \beta_{0i} + \beta_1 x_{ij1} + \beta_2 x_{ij2} + \ldots + \beta_m x_{ijm} + \varepsilon_{ij} \phantom{...} i = 1,2,\ldots,\mathit{ngroup}; \phantom{...} j = 1,2,\ldots,n_i\]

where the observed value of \(y_{ij}\) constitutes the j-th response in the i-th group, \(\beta_{0i}\) denotes the y intercept for the regression function for the i-th group, \(\beta_1,\beta_2,\ldots,\beta_m\) are the regression coefficients for the covariates, and the \(\varepsilon_{ij}\) ’s are independently distributed normal errors with mean zero and variance \(\sigma^2\). This model allows the regression function for each group to have different intercepts. However, the remaining m regression coefficients are the same for each group, i.e., the regression functions are parallel.

In practice, sometimes the regression functions are not parallel. In addition to estimates for the model assuming parallelism, ancovar computes estimates and summary statistics for the separate regressions for each group. These estimates can be examined using the optional arguments coefTables and regAnova.

Estimates for the \(\beta_{0i}\)’s and \(\beta_1\), \(\beta_2\), …, \(\beta_m\) in the model assuming parallelism are returned using the optional argument coef. Summary statistics are also computed for this model.

The adjusted group means, stored in the last column of xymean, are computed using the formula:

\[\hat{\beta}_{oi} + \hat{\beta}_1 \overline{x}_1 + \hat{\beta}_2 \overline{x}_2 + \ldots + \hat{\beta}_{n\mathrm{cov}} \overline{x}_{n\mathrm{cov}}\]

The estimated covariance between the \(i_1\)-th and \(i_2\)-th adjusted group mean is given by

\[v_{i_1i_2} + \sum_{r=1}^{m} \sum_{s=1}^{m} \overline{x}_r v_{k+r, k+s} \overline{x}_s + \sum_{r=1}^{m} \overline{x}_r v_{i_1, k+r} + \sum_{r=1}^{m} \overline{x}_r v_{i_2, k+r}\]

where \(v_{pq}\) is the entry in covCoef[(p - 1)(ngroup + ncov) + q -1] and is the estimated covariance between the p‑th and q‑th estimated coefficients in the regression function.

A discussion of formulas and interpretations for the one-way analysis of covariance problem appears in most elementary statistics texts, e.g., Snedecor and Cochran (1967, Chapter 14).

Examples¶

Example 1¶

This example fits a one-way analysis of covariance model assuming parallelism using data discussed by Snedecor and Cochran (Table 14.6.1, pages 432−436). The responses are concentrations of cholesterol (in mg/100 ml) in the blood of two groups of women: women from Iowa and women from Nebraska. Age of a woman is the single covariate. The cholesterol concentrations and ages of the women according to state are shown in the following table. (There are 11 Iowa women and 19 Nebraska women in the study. Only the first 5 women from each state are shown here.)

Iowa		Nebraska
Age	Cholesterol	Age	Cholesterol
46	181	18	137
52	228	44	173
39	182	33	177
65	249	78	241
54	259	51	225

There is no evidence from the data to indicate that the regression lines for cholesterol concentration as a function of age are not parallel for Iowa and Nebraska women (p-value is 0.5425). The parallel line model suggests that Nebraska women may have higher cholesterol concentrations than Iowa women. The cholesterol concentrations (adjusted for age) are 195.5 for Iowa women versus 224.2 for Nebraska women. The difference is 28.7 with an estimated standard error of

\[\sqrt{170.4 + 97.4 - 2(2.9)} = 16.1\]

from __future__ import print_function
from numpy import *
from pyimsl.stat.ancovar import ancovar
from pyimsl.stat.writeMatrix import writeMatrix

ncov = 1
ngroup = 2
ni = [11, 19]
testpl = []
aov = []
xymean = []
covm = []
y = [181.0, 228.0, 182.0, 249.0, 259.0,
     201.0, 121.0, 339.0, 224.0, 112.0,
     189.0, 137.0, 173.0, 177.0, 241.0,
     225.0, 223.0, 190.0, 257.0, 337.0,
     189.0, 214.0, 140.0, 196.0, 262.0,
     261.0, 356.0, 159.0, 191.0, 197.0]
x = [46.0, 52.0, 39.0, 65.0, 54.0,
     33.0, 49.0, 76.0, 71.0, 41.0,
     58.0, 18.0, 44.0, 33.0, 78.0,
     51.0, 43.0, 44.0, 58.0, 63.0,
     19.0, 42.0, 30.0, 47.0, 58.0,
     70.0, 67.0, 31.0, 21.0, 56.0]

aov = ancovar(ni, y, x,
              parallelTests=testpl,
              xymean=xymean,
              covMeans=covm)

print("             * * * ANALYSIS OF VARIANCE * * * ")
print("                  Sum of         Mean                Prob of")
print("Source   DF      Squares        Square    Overall F  Larger F")
print("Model   %3.0f   %10.2f     %9.2f      %2.2f    %8.6f" %
      (aov[0], aov[3], aov[6], aov[8], aov[9]))
print("Error   %3.0f   %10.2f     %9.2f      " % (aov[1], aov[4], aov[7]))
print("Total   %3.0f   %10.2f  " % (aov[2], aov[5]))
print("")

print("             * * * TEST FOR PARALLELISM  * * * ")
print("                       Sum of     Mean         F     Prob of")
print("SOURCE           DF    Squares   Square       TEST   Larger F")
print("Extra due to")
print("Nonparallelism %3.0f %10.2f    %7.2f    %7.5f   %5.4f" %
      (testpl[0], testpl[3], testpl[6], testpl[8], testpl[9]))
print("Extra Assuming")
print("Nonparallelism %3.0f %10.2f    %7.2f " %
      (testpl[1], testpl[4], testpl[7]))
print("Error Assuming")
print("Parallelism    %3.0f %10.2f   " % (testpl[2], testpl[5]))

writeMatrix("XY Mean Matrix", xymean)
writeMatrix("Var./Covar. Matrix of Adjusted Group Means", covm)

Output¶

             * * * ANALYSIS OF VARIANCE * * * 
                  Sum of         Mean                Prob of
Source   DF      Squares        Square    Overall F  Larger F
Model     2     54432.75      27216.38      14.97    0.000042
Error    27     49103.91       1818.66      
Total    29    103536.67  

             * * * TEST FOR PARALLELISM  * * * 
                       Sum of     Mean         F     Prob of
SOURCE           DF    Squares   Square       TEST   Larger F
Extra due to
Nonparallelism   1     709.05     709.05    0.38093   0.5425
Extra Assuming
Nonparallelism  26   48394.86    1861.34 
Error Assuming
Parallelism     27   49103.91   
 
                   XY Mean Matrix
             1            2            3            4
1         11.0         53.1        207.7        195.5
2         19.0         45.9        217.1        224.2
3         30.0         48.6        213.7        213.7
 
Var./Covar. Matrix of Adjusted Group Means
                     1            2
        1        170.4         -2.9
        2         -2.9         97.4

Figure 4.3 — Plot of Cholesterol Concentrations and Fitted Parallel Lines by State

Example 2¶

This example fits a one-way analysis of covariance model and performs a test for parallelism using data discussed by Snedecor and Cochran (1967, Table 14.8.1, pages 438-443). The responses are weight gains (in pounds per day) of 40 pigs for four groups of pigs under varying treatments. Two covariates-initial age (in days) and initial weight (in pounds) are used. For each treatment, there are 10 pigs. Only the first five pigs from each treatment are shown here.

Treatment 1			Treatment 2			Treatment 3			Treatment 4
Age	Wt.	Gain	Age	Wt.	Gain	Age	Wt.	Gain	Age	Wt.	Gain
78	61	1.40	78	74	1.61	78	80	1.67	77	62	1.40
90	59	1.79	99	75	1.31	83	61	1.41	71	55	1.47
94	76	1.72	80	64	1.12	79	62	1.73	78	62	1.37
71	50	1.47	75	48	1.35	70	47	1.23	70	43	1.15
99	61	1.26	94	62	1.29	85	59	1.49	95	57	1.22

For these data, the test for non-parallelism is not statistically significant (\(p=0.901\)). The one-way analysis of covariance test for the treatment means adjusted for the covariates, assuming parallel slopes, is statistically significant at a stated significance level of \(\alpha=0.05\), (\(p=0.04931\)).

Multiple comparisons can be done using the least significant difference approach of comparing each pair of treatment groups with the two-sample student-t test. Since the adjusted means in the one-way analysis of covariance are correlated, the standard error for these comparisons must be computed using the variances and covariances in covMeans. The standard errors for these comparisons are fairly similar ranging from 0.0630 to 0.0638. The Student’s t comparisons identify differences between groups 1 and 2, and 1 and 4 as being statistically significant with p-values of 0.01225 and 0.03854 respectively.

from __future__ import print_function
from numpy import *
from pyimsl.stat.ancovar import ancovar
from pyimsl.stat.writeMatrix import writeMatrix
from pyimsl.stat.tCdf import tCdf

ncov = 2
ngroup = 4
nobs = 40
ni = [10, 10, 10, 10]
aov = []
testpl = []
adj_aov = []
xymean = []
covm = []
x1 = [78.0, 90.0, 94.0, 71.0, 99.0, 80.0, 83.0, 75.0, 62.0, 67.0,
      78.0, 99.0, 80.0, 75.0, 94.0, 91.0, 75.0, 63.0, 62.0, 67.0,
      78.0, 83.0, 79.0, 70.0, 85.0, 83.0, 71.0, 66.0, 67.0, 67.0,
      77.0, 71.0, 78.0, 70.0, 95.0, 96.0, 71.0, 63.0, 62.0, 67.0]
x2 = [61.0, 59.0, 76.0, 50.0, 61.0, 54.0, 57.0, 45.0, 41.0, 40.0,
      74.0, 75.0, 64.0, 48.0, 62.0, 42.0, 52.0, 43.0, 50.0, 40.0,
      80.0, 61.0, 62.0, 47.0, 59.0, 42.0, 47.0, 42.0, 40.0, 40.0,
      62.0, 55.0, 62.0, 43.0, 57.0, 51.0, 41.0, 40.0, 45.0, 39.0]
y = [1.40, 1.79, 1.72, 1.47, 1.26, 1.28, 1.34, 1.55, 1.57, 1.26,
     1.61, 1.31, 1.12, 1.35, 1.29, 1.24, 1.29, 1.43, 1.29, 1.26,
     1.67, 1.41, 1.73, 1.23, 1.49, 1.22, 1.39, 1.39, 1.56, 1.36,
     1.40, 1.47, 1.37, 1.15, 1.22, 1.48, 1.31, 1.27, 1.22, 1.36]
#
# setup covariate input matrix
#
x = empty((40, 2), dtype=double)
for i in range(0, nobs):
    x[i, 0] = x1[i]
    x[i, 1] = x2[i]

aov = ancovar(ni, y, x,
              parallelTests=testpl,
              adjAnova=adj_aov,
              xymean=xymean,
              covMeans=covm)

print("")
print("             * * * TEST FOR PARALLELISM  * * * ")
print("                       Sum of     Mean         F     Prob of")
print("SOURCE           DF    Squares   Square       TEST   Larger F")
print("Extra due to")
print("Nonparallelism %3.0f %10.2f    %7.2f    %7.5f   %5.3f"
      % (testpl[0], testpl[3], testpl[6], testpl[8], testpl[9]))
print("Extra Assuming")
print("Nonparallelism %3.0f %10.2f    %7.2f " %
      (testpl[1], testpl[4], testpl[7]))
print("Error Assuming")
print("Parallelism    %3.0f %10.2f   " % (testpl[2], testpl[5]))

print("")
print("             * * * ANALYSIS OF VARIANCE * * * ")
print("                  Sum of         Mean                Prob of")
print("Source   DF      Squares        Square    Overall F  Larger F")
print("Model   %3.0f      %f     %f    %f    %5.4f"
      % (aov[0], aov[3], aov[6], aov[8], aov[9]))
print("Error   %3.0f      %f     %f      "
      % (aov[1], aov[4], aov[7]))
print("Total   %3.0f      %f  " % (aov[2], aov[5]))

print("")
print("         * * * ADJUSTED ANALYSIS OF VARIANCE  * * * ")
print("                                Sum of        F     Prob of")
print("Source                    DF    Squares      TEST   Larger F")
print("Groups after Covariates %3.0f   %10.2f    %5.2f    %7.5f"
      % (adj_aov[0], adj_aov[2], adj_aov[4], adj_aov[6]))
print("Covariates after Groups %3.0f   %10.2f    %5.2f    %7.5f" %
      (adj_aov[1], adj_aov[3], adj_aov[5], adj_aov[7]))

print("           * * * GROUP MEANS * * * ")
print("GROUP  | Unadjusted   |  Adjusted |  Std. Error")

for i in range(0, ngroup):
    stderr = sqrt(covm[i][i])
    print("  %d    |   %5.4f     |   %5.4f  |   %7.4f" %
          (i + 1, xymean[i][ngroup - 1], xymean[i][ngroup], stderr))

print("      * * * STUDENT-T MULTIPLE COMPARISONS * * * ")
print(" GROUPS  |    DIFF   | Std. Error | Student-t | P-Value")
for i in range(0, ngroup):
    for j in range(i + 1, ngroup):
        delta = xymean[i][ngroup] - xymean[j][ngroup]
        stderr = sqrt(covm[i][i] + covm[j][j] - 2.0 * covm[i][j])
        t = delta / stderr
        df = xymean[i][0] + xymean[j][0] - 2
        pvalue = 1.0 - tCdf(t, df)
        print(" %d vs %d  |  %7.4f  |  %7.4f   | %7.3f   | %7.5f"
              % (i + 1, j + 1, delta, stderr, t, pvalue))

Output¶

             * * * TEST FOR PARALLELISM  * * * 
                       Sum of     Mean         F     Prob of
SOURCE           DF    Squares   Square       TEST   Larger F
Extra due to
Nonparallelism   6       0.05       0.01    0.35534   0.901
Extra Assuming
Nonparallelism  28       0.62       0.02 
Error Assuming
Parallelism     34       0.67   

             * * * ANALYSIS OF VARIANCE * * * 
                  Sum of         Mean                Prob of
Source   DF      Squares        Square    Overall F  Larger F
Model     5      0.352517     0.070503    3.576390    0.0105
Error    34      0.670261     0.019714      
Total    39      1.022777  

         * * * ADJUSTED ANALYSIS OF VARIANCE  * * * 
                                Sum of        F     Prob of
Source                    DF    Squares      TEST   Larger F
Groups after Covariates   3         0.17     2.90    0.04931
Covariates after Groups   2         0.17     4.44    0.01939
           * * * GROUP MEANS * * * 
GROUP  | Unadjusted   |  Adjusted |  Std. Error
  1    |   1.4640     |   1.4614  |    0.0448
  2    |   1.3190     |   1.3068  |    0.0446
  3    |   1.4450     |   1.4429  |    0.0447
  4    |   1.3250     |   1.3418  |    0.0449
      * * * STUDENT-T MULTIPLE COMPARISONS * * * 
 GROUPS  |    DIFF   | Std. Error | Student-t | P-Value
 1 vs 2  |   0.1546  |   0.0630   |   2.455   | 0.01225
 1 vs 3  |   0.0185  |   0.0637   |   0.290   | 0.38750
 1 vs 4  |   0.1196  |   0.0638   |   1.875   | 0.03854
 2 vs 3  |  -0.1362  |   0.0632   |  -2.153   | 0.97743
 2 vs 4  |  -0.0350  |   0.0638   |  -0.549   | 0.70528
 3 vs 4  |   0.1011  |   0.0631   |   1.602   | 0.06330