ONE WAY ANOVA

ONE WAY ANOVA

Analysis Command

Perform a One Way Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a data analysis technique for examining the significance of the factors (= independent variables) in a multi-factor model. Each factor has a certain number of values it can have (these are referred to as the levels of a factor). The number of levels can vary betweeen factors. Each factor and level combination is a cell. Balanced designs are those in which each cell has an equal number of observations and unbalanced designs are those in which the number of observations can vary among cells. A treatment is defined as a single combination of the levels of the factors. For the one-factor case, a treatment is equivalent to a level.

The basic ANOVA model for one factor is

\( y_{ij} = \mu + \tau_{i} + \epsilon_{ij} \hspace{0.5in} i = 1, \ldots, k, j = 1, \ldots, n_{i} \)

where

\( y_{ij} \)	=	the j-th observation of the i-th treatment
\( \mu \)	=	the population (grand) mean
\( \tau_{i} \)	=	the effect of the i-th treatment
\( \epsilon_{ij} \)	=	the error for the j-th observation of the i-th treatment
\( n_{i} \)	=	the number of observations for treatment \( i \)
\( k \)	=	the number of treatments

The errors are assumed to be independent and normally distributed with mean 0 and standard deviation \( \sigma \) and \( \mu \) is assumed to be a fixed parameter.

In the fixed effects ANOVA model, the levels of the factor are assumed to be fixed and the \( \tau_{i} \) are also assumed to be fixed parameters. In the random effects ANOVA model, the levels of a factor are assumed to be a random sample from a population of possible level values and the \( \tau_{i} \) are assumed to be random variables that are normally distributed with mean zero and standard deviation \( \sigma_{\tau} \). The \( \tau_{i} \) and \( \epsilon_{ij} \) are assumed to be independent for all \( i \) and \( j \). Although the basic ANOVA calculations are the same for the fixed and random effects model, the hypothesis being tested is different. In a fixed effects model, ANOVA is used to test the hypothesis that \( \tau_1 = \tau_2 = ... = \tau_k \). In a random effects model, ANOVA is used to test the hypothesis that \( \sigma_{\tau} = 0 \). Note that testing for the equality of the \( \tau_{i} \) is not meaningful in the context of a random effects model.

For example, suppose one factor is the operator of a machine. In a fixed effects model, we would be testing whether the average performance varies between the specific operators tested. For a random effects model, we assume that the specific operators tested were randomly selected from a larger pool of available operators. In this case, we are not interested in the average performance of the specific operators tested. Rather we are interested in the variability between operators.

The basic computations for the one factor fixed effects ANOVA are given here (and in most introductory statistics text books)

https://www.itl.nist.gov/div898/handbook/prc/section4/prc431.htm https://www.itl.nist.gov/div898/handbook/prc/section4/prc432.htm https://www.itl.nist.gov/div898/handbook/prc/section4/prc433.htm

Basically, the sum of squares are decomposed into a "treatment" sum of squares and an "error" sum of squares. Essentially we are comparing the ratio of the between treatment variability and the within treatment variability. For a fixed effects case, if the between variability is small compared to the within variability we conclude the treatment means are essentially equal. Similarly for the random effects model, if the between variability is small compared to the within variability we conclude the variance between treatments is essentially zero.

The Dataplot ANOVA command performs an analysis of variance for up to 10 factors. However, this command is intended for the fixed effects case with balanced data (i.e., all cells have the same sample size).

The ONE WAY ANOVA command specifically handles the case of a single factor variable. However, it handles a broader range cases than the regular ANOVA command and provides some additional output. Specifically,

1. The ONE WAY ANOVA command allows summary data (i.e., the mean, standard deviation and sample size for each level of the factor variable) in addition to raw data.

2. The ONE WAY ANOVA command allows a random effects model to be specified. This case will generate a variance components table (more will be said about this in the Notes section below).

3. For the fixed effects case, the ONE WAY ANOVA command generates several additional tables. Specifically
   • A table containing confidence intervals for the treatment means is generated.

   • A table containg confidence intervals for the difference of the treatment means is generated.

   • A table containing Bonferroni adjusted confidence intervals for the difference of treatment means is generated.

   • A table containing Tukey-Kramer adjusted confidence intervals for the difference of treatment means is generated.

   Details about these tables is given in the Notes section below.

   Since random effect models are not concerned with the mean values and difference of the mean values for the specific treatments, these tables will not be generated for that case.

4. The ONE WAY ANOVA command supports unbalanced data.

   There are two warnings that should be noted when dealing with unbalanced data. The balanced ANOVA case is known to be somewhat robust to the assumption of equal variances for the treatments. A common rule of thumb is that if the ratio of the largest treatment variance to the smallest treatment variance is less than 2, then the ANOVA should be robust to unequal variances. The unbalanced case is more sensitive to the equal variances assumption (and there is no corresponding rule of thumb in this case). In addition, the unbalanced ANOVA is known to be less powerful than the balanced ANOVA case. By less powerful, we mean that it takes larger differences in the treatment means to detect a statistically significant difference.

ONE WAY ANOVA <y> <x>       <SUBSET/EXCEPT/FOR qualification>
where <y> is the response (= dependent) variable;
            <x> is the factor (= independent) variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax generates a fixed effects ANOVA based on raw data.

ONE WAY SUMMARY ANOVA <ymean> <ysd> <yn>
                        <SUBSET/EXCEPT/FOR qualification>
where <ymean> is a variable containing the treatment means;
            <ysd> is a variable containing the treatment standard deviations;
            <yn> is a variable containing the treatment sample sizes;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax generates a fixed effects ANOVA based on summary data.

ONE WAY RANDOM EFFECTS ANOVA <y> <x>
                        <SUBSET/EXCEPT/FOR qualification>
where <y> is the response (= dependent) variable;
            <x> is the factor (= independent) variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax generates a random effects ANOVA based on raw data.

ONE WAY SUMMARY RANDOM EFFECTS ANOVA
                        <ymean> <ysd> <yn>
                        <SUBSET/EXCEPT/FOR qualification>
where <ymean> is a variable containing the treatment means;
            <ysd> is a variable containing the treatment standard deviations;
            <yn> is a variable containing the treatment sample sizes;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax generates a random effects ANOVA based on summary data.

ONE WAY ANOVA Y X
ONE WAY SUMMARY ANOVA YMEAN YSD YN
ONE WAY RANDOM EFFECTS ANOVA Y X
ONE WAY SUMMARY RANDOM EFFECTS ANOVA YMEAN YSD YN
ONE WAY ANOVA Y X SUBSET Y > 0
ONE WAY ANOVA Y X SUBSET X = 1 TO 6

The ONE WAY ANOVA command saves the residuals in the variable RES and the predicted values in the variable PRED. These can be used to generate various diagnostic plots.

The predicted values are the estimated grand mean plus the estimated treatment effect. The residuals are the response variable minus the predicted values.

The PRED and RES variables will not be saved if summary data is used.

For the random effects model, a variance component table will be printed. An example table is shown in the Program 2 example below.

The variance component table is used to specify the percentage of the variance that can be attributed to the treatment (between) and how much to the error (within).

The variance component for the treatment is computed as

\( s_{Treatment}^{2} = \max(0,\frac{MSTR - MSE}{n0}) \)

where MSTR denotes the mean square of the treatment and MSE denotes the mean square error and

\( n_{0} = \frac{1}{k-1} \left(\sum_{i=1}^{k}{n_{i}} - \frac{\sum_{i=1}^{k}{n_{i}^{2}}} {\sum_{i=1}^{k}{n_{i}}}\right) \)

The MSTR and MSE values are obtained from the standard ANOVA table. In the balanced case, n0 is equal to the level sample size. The variance component for the error is simply the MSE.

The variance components are often given as a proportion of the total variance. In this case, the variance component is divided by the sum of the treatment variance component and the error variance component. This is also referred to as the intraclass correlation coefficient. Dataplot prints this expressed as a percentage rather than a proportion.

The confidence interval for the treatment variance component is computed as

\( \mbox{Lower CL} = \mbox{SSTR} \frac{\left(1 - \frac{F_u}{F_0}\right)} {n_0 \chi^{2}_{(\alpha/2,k-1)}} \)

\( \mbox{Upper CL} = \mbox{SSTR} \frac{\left(1 - \frac{F_l}{F_0}\right)} {n_0 \chi^{2}_{(1-(\alpha/2),k-1)}} \)

where

\( \chi^{2} \)	=	the chi-square percent point function
\( F_l \)	=	\( F_{(\alpha/2,k-1,ntotal-k)} \)
\( F_u \)	=	\( F_{(1-(\alpha/2),k-1,ntotal-k)} \)
\( F_0 \)	=	F statistic from the ANOVA
F	=	the F percent point function
ntotal	=	the total number of observations

The confidence interval for the error variance component is computed as

\( \mbox{Lower CL} = (ntotal-k) \frac{\mbox{MSE}} {\chi^{2}_{(\alpha/2,ntotal-k)}} \)

\( \mbox{Upper CL} = (ntotal-k) \frac{\mbox{MSE}} {\chi^{2}_{(1-(\alpha/2),ntotal-k)}} \)

Approximate confidence intervals for the intraclass correlation coefficient are computed as

\( \mbox{Lower CL} = \frac{\mbox{L}} {1 + \mbox{L}} \)

\( \mbox{Upper CL} = \frac{\mbox{U}} {1 + \mbox{U}} \)

where

\( \mbox{L} = \frac{1}{n_0} (\frac{F_{0}}{F_{u}} - 1) \)

\( \mbox{U} = \frac{1}{n_0} (\frac{F_{0}}{F_{l}} - 1) \)

where \( F_0 \), \( F_l \) and \( F_u \) are as defined above.

For the fixed effects model, confidence intervals for the level means are computed as follows
\( \bar{y_{i}} \pm t_{(1-(\alpha/2),ntotal-k)} \sqrt{\frac{\mbox{MSE}} {n_{i}}} \)

with \( \bar{y_{i}} \) denoting the mean of the i-th level and t denoting the t percent point function.

If the ANOVA shows a statistically significant difference in the level means, a follow-up step is to compare the pairwise difference in treatment means to determine which differences are statistically significant. This is a subset of possible multiple comparisons. There are a number of different ways to perform these pairwise comparisons.

The first table generates confidence intervals based on Fisher's Least Protected Difference (LSD)

\( |\bar{y_{i}} - \bar{y_{j}}| \pm t_{(1-(\alpha/2),ntotal-k)} \sqrt{\frac{\mbox{MSE}} {\frac{1}{n_i} + \frac{1}{n_j}}} \)

This is just the standard confidence interval for the difference between means. The protected part is based on the idea that these differences are only analyzed if the F test from the ANOVA is statistically significant. The last column in the table provides p-values (p-values less than alpha indicate statistical significance).

The second table generates confidence intervals based on Bonferroni's adjustment to alpha. The LSD method provides pairwise statistical signficance, but does not provide simultaneous statistical significance. Bonferroni addresses this with the following correction to alpha

\( \alpha^{'} = \frac{\alpha}{k(k-1)/2} \)

The table header prints the adjusted alpha. This can be compared to the p-values in the last column of the table to determine statistical significance.

The third table generates Tukey-Kramer honest significant difference (HSD) confidence intervals. These confidence intervals are based on

\( |\bar{y_{i}} - \bar{y_{j}}| \pm q_{(\alpha,k,ntotal-k)} \sqrt{\frac{\mbox{MSE}} {\frac{1}{n_i} + \frac{1}{n_j}}} \)

where q denotes the studentized range percent point function. Like the Bonferroni intervals, these intervals provide simultaneous statistical significance.

The generated tables are based on pairwised differences between level means. More general comparisons are based on contrasts and linear combnations. A contrast is a linear combination of two or more factor level means with coefficients that sum to zero

\( C = c_{1} \mu_{1} + c_{2} \mu_{2} + \ldots + c_{r} \mu_{r} \)

with \( c_{1}, c_{2}, \dots , c_{r} \) denoting the contrast coefficients and \( \mu_{1}, \mu_{2}, \dots , \mu_{r} \) denoting the factor level means and where

\( \sum_{i=1}^{r}{c_{i}} = 0 \)

For example, the pairwise difference betweem two level means has coefficients \( c_1 \) = 1 and \( c_2 \)= -1. To compare levels 1 and 2 with level 3, the coefficients are \( c_1 \) = 1, \( c_2 \)= 1, and \( c_3 \) = -2.

Two contrasts are orthogonal if the sum of the products of corresponding coefficients (i.e., coefficients for the same means) also sum to zero. That is

\( \sum_{i=1}^{r}{c_{i} d_{i}} = 0 \)

where \( c \) and \( d \) are coefficients for two contrasts.

For the unbalanced case, this is

\( \sum_{i=1}^{r}{n_{i} c_{i} d_{i}} = 0 \)

A linear combination is similar to a contrast, but it does not require that the coefficients sum to zero.

A contrast is estimated by

\( \hat{C} = \sum_{i=1}^{r}{c_{i} \bar{y_{i}}} \)

The variance of the contrast is

\( \mbox{Var}(\hat{C}) = \sigma^{2} \sum_{i=1}^{r}{\frac{c_{i}^{2}} {n_{i}}} \)

and a confidence interval is given by

\( \hat{C} \pm t_{(1-(\alpha/2),ntotal-r)} \sqrt{\mbox{Var}(\hat{C})} \)

Estimates and confidence intervals for linear combinations are generated using the same formulas as contrasts.

Montgomery (pp. 60-62) shows how the treatment sum of squares can be partitioned into contrast sum of squares for orthogonal contrasts. Note that constrasts should be chosen before the data is collected.

By default, an \( \alpha \) = 0.05 is used in the above formulas. To use a different value of alpha, enter
LET ALPHA = <value>

Typical values for alpha are 0.01, 0.05 and 0.10.

The following quantities are written to files
1. dpst1f.dat - the level id, the level mean, the effect estimate and the standard deviation of the effect estimate

2. dpst2f.dat - the level id, the level mean, the level standard deviation, and the lower and upper confidence limits for the level mean

3. dpst3f.dat - the first level id, the second level id, the Fisher LSD lower and upper confidence limits and the p-value for the difference of means

4. dpst4f.dat - the first level id, the second level id, the Bonferroni lower and upper confidence limits and the p-value for the difference of means

5. dpst5f.dat - the first level id, the second level id, the Tukey-Kramer standard error, the Tukey-Kramer lower and upper confidence limits and the 90%, 95% and 99% critical values for Tukey-Kramer

The following parameters are saved after the ONE WAY ANOVA command is performed.

STATVAL	-	value of the F test statistic
STATCDF	-	CDF of the F test statistic
PVALUE	-	p-value of the F test statistic
MSTR	-	treatment mean square
MSE	-	error mean square
SSTR	-	treatment sum of squares
SSE	-	error sum of squares
SSTOT	-	total sum of squares

None

ANALYSIS OF VARIANCE is a synonym for ANOVA
1 WAY is a synonym for ONE WAY
RANDOM EFFECTS SUMMARY is a synonym for SUMMARY RANDOM EFFECTS

ANOVA	=	Perform a fixed effects ANOVA for balanced data (up to 10 factors)
KRUSKAL WALLIS	=	Perform a Kruskal Wallis test.
MEDIAN POLISH	=	Carries out a robust ANOVA.
YATES ANALYSIS	=	Analyze a Yate's design.
BLOCK PLOT	=	Generate a block plot.
DEX SCATTER PLOT	=	Generates a dex scatter plot.
DEX ... PLOT	=	Generates a dex plot for a statistic.
DEX ... EFFECTS PLOT	=	Generates a dex effects plot for a statistic.
T TEST	=	Carries out a t test.
PLOT	=	Plots (e.g., residuals and GANOVA ).

One way analysis of variance is covered in most introductory statistics books and more advanced books covering linear models.

Neter, Wasserman and Kunter (1990), "Applied Linear Statistical Models," 3rd ed., Irwin.

Searle, Casella and McCulloch (1992), "Variance Components," Wiley, Chapter 3.

Montgomery (1976), "Design and Analysis of Experiments," Second Edition, Wiley, chapters 3 and 4.

Analysis of Variance

2023/09

 
. Fixed Effects Case
.
. Step 1:   Create the data
.
read x y
1   6.9
1   5.4
1   5.8
1   4.6
1   4.0
2   8.3
2   6.8
2   7.8
2   9.2
2   6.5
3   8.0
3  10.5
3   8.1
3   6.9
3   9.3
4   5.8
4   3.8
4   6.1
4   5.6
4   6.2
end of data
.
set let cross tabulate collapse
let ymean = cross tabulate mean y x
let ysd   = cross tabulate sd   y x
let yn    = cross tabulate size x
.
. Step 2:   Run standard anova command and summary anova command
.
echo on
capture 1way.out
anova y x
one way summary anova ymean ysd yn
one way         anova y x
end of capture
echo off
    

The following output is generated in ONEWAYAN.OUT

       *****************
       **  anova y x  **
       *****************
  
  
              1-Way Fixed Effects Analysis of Variance--Balanced Case
  
 Response Variable: Y
 Factor  1 Variable: X
  
 Summary Statistics:
 Number of Observations:                              20
 Number of Factors:                                    1
 Number of Levels for Factor  1:                       4
 Number of Distinct Cells:                             4
  
 Grand Mean:                                     6.78000
 Grand Standard Deviation:                       1.77870
  
 Residual Standard Deviation:                    1.15358
 Residual Degrees of Freedom:                         16
  
 Replication Case:
 Replication Standard Deviation:                 1.15358
 Replication Degrees of Freedom:                      16
 Lack of Fit F Test cannot be done
 because there are 0 degrees of freedom
 in the numerator of the F ratio.  This
 happens when the number of parameters
 fitted is identical to the number of
 distinct subsets.
  
  
             ANOVA Table
  
 --------------------------------------------------------------------------------
             Source  DF Sum of Squares    Mean Square    F Statistic    F CDF Sig
 --------------------------------------------------------------------------------
  Total (Corrected)  19       60.11200        3.16379
 --------------------------------------------------------------------------------
          Factor  1   3       38.82000       12.94000         9.7238  99.932%  **
 --------------------------------------------------------------------------------
           Residual  16       21.29200        1.33075
  
  
             Estimation
  
 ------------------------------------------------------------
            Level-ID      NI       Mean     Effect SD(Effect)
 ------------------------------------------------------------
  Factor  1        1       5    5.34000   -1.44000    0.44678
                   2       5    7.72000    0.94000    0.44678
                   3       5    8.56000    1.78000    0.44678
                   4       5    5.50000   -1.28000    0.44678
  
  
             Models
  
 ---------------------------------------------------------
 Model                         Residual Standard Deviation
 ---------------------------------------------------------
 Constant               Only--  1.77870
 Constant and Factor  1 Only--  1.15358
 Constant and All  2 Factors--  1.15358
  
  
       ******************************************
       **  one way summary anova ymean ysd yn  **
       ******************************************
  
  
             One-Way Fixed Effects Analysis of Variance: Summary Data
                  H0: Effect(1) = Effect(2) = ... - Effect(k) = 0
  
 Means Variable:       YMEAN
 SD Variable:          YSD
 Sample Size Variable: YN
  
 Summary Statistics:
 Total Number of Observations                         20
 Number of Levels:                                     4
 Grand Mean:                                     6.78000
  
  
  
             ANOVA Table: Fixed Effects
  
 --------------------------------------------------------------------------------------------------
               Source   DF Sum of Squares    Mean Square    F Statistic   F CDF (%)     P-Value Sig
 --------------------------------------------------------------------------------------------------
    Total (Corrected)   19       60.11200
 --------------------------------------------------------------------------------------------------
  Treatment (Between)    3       38.82000       12.94000         9.7238     99.9316    0.000684  **
    Residual (Within)   16       21.29200        1.33075
  
  
             Estimation: Grand Mean =   0.6780000E+01
  
 -------------------------------------------------------
 Level
    ID N(i)           Mean         Effect     SD(Effect)
 -------------------------------------------------------
     1    5        5.34000       -1.44000        0.44678
     2    5        7.72000        0.94000        0.44678
     3    5        8.56000        1.78000        0.44678
     4    5        5.50000       -1.28000        0.44678
  
  
             95% Confidence Intervals for Treatment Means
  
 --------------------------------------------------------------------------------
                                          Treatment          Lower          Upper
            Treatment      Treatment         Sample     Confidence     Confidence
     I           Mean             SD           Size          Limit          Limit
 --------------------------------------------------------------------------------
     1        5.34000        1.11714              5        4.43930        6.24070
     2        7.72000        1.10318              5        6.81930        8.62070
     3        8.56000        1.37768              5        7.65930        9.46070
     4        5.50000        0.97980              5        4.59930        6.40070
  
  
             95% Confidence Intervals for the Difference of Treatment Means
  
 ----------------------------------------------------------------------
             Difference of          Lower          Upper         Fisher
                 Treatment     Confidence     Confidence            LSD
     I    J          Means          Limit          Limit        P-Value
 ----------------------------------------------------------------------
     1    2       -2.38000       -3.65378       -1.10622        0.00489
     1    3       -3.22000       -4.49378       -1.94622        0.00043
     1    4       -0.16000       -1.43378        1.11378        0.82919
     2    3       -0.84000       -2.11378        0.43378        0.26651
     2    4        2.22000        0.94622        3.49378        0.00775
     3    4        3.06000        1.78622        4.33378        0.00069
  
  
             95% Bonferroni Intervals for the Difference of Treatment Means
  
 Bonferroni Adjusted Alpha = 0.0083
 ----------------------------------------------------------------------
     I    J  Difference of          Lower          Upper
                 Treatment     Confidence     Confidence
                     Means          Limit          Limit        P-Value
 ----------------------------------------------------------------------
     1    2       -2.38000       -4.33021       -0.42979        0.00489
     1    3       -3.22000       -5.17021       -1.26979        0.00043
     1    4       -0.16000       -2.11021        1.79021        0.82919
     2    3       -0.84000       -2.79021        1.11021        0.26651
     2    4        2.22000        0.26979        4.17021        0.00775
     3    4        3.06000        1.10979        5.01021        0.00069
  
  
             Tukey-Kramer Honest Significant Difference (HSD)
  
 Multiple Comparisons: alpha = 0.05
 ----------------------------------------------------------------------
              |Difference|                         Lower          Upper
              of Treatment       Standard     Confidence     Confidence
     I    J          Means          Error          Limit          Limit
 ----------------------------------------------------------------------
     1    2        2.38000        0.51590        0.28038        4.47962
     1    3        3.22000        0.51590        1.12038        5.31962
     1    4        0.16000        0.51590       -1.93962        2.25962
     2    3        0.84000        0.51590       -1.25962        2.93962
     2    4        2.22000        0.51590        0.12038        4.31962
     3    4        3.06000        0.51590        0.96038        5.15962
  
  
       *********************************
       **  one way         anova y x  **
       *********************************
  
  
             One-Way Fixed Effects Analysis of Variance: Raw Data
               H0: Effect(1) = Effect(2) = ... - Effect(k) = 0
  
 Group-ID Variable:    Y
  
 Summary Statistics:
 Total Number of Observations                         20
 Number of Levels:                                     4
 Grand Mean:                                     6.78000
 Grand Standard Deviation:                       1.77870
  
  
  
             ANOVA Table: Fixed Effects
  
 --------------------------------------------------------------------------------------------------
               Source   DF Sum of Squares    Mean Square    F Statistic   F CDF (%)     P-Value Sig
 --------------------------------------------------------------------------------------------------
    Total (Corrected)   19       60.11200
 --------------------------------------------------------------------------------------------------
  Treatment (Between)    3       38.82000       12.94000         9.7238     99.9316    0.000684  **
    Residual (Within)   16       21.29200        1.33075
  
  
             Estimation: Grand Mean =   0.6780000E+01
  
 -------------------------------------------------------
 Level
    ID N(i)           Mean         Effect     SD(Effect)
 -------------------------------------------------------
     1    5        5.34000       -1.44000        0.44678
     2    5        7.72000        0.94000        0.44678
     3    5        8.56000        1.78000        0.44678
     4    5        5.50000       -1.28000        0.44678
  
  
             Models
  
 -------------------------------------------------------
 Model                       Residual Standard Deviation
 -------------------------------------------------------
 Constant                                        1.77870
 Constant + Factor                               1.15358
  
  
             95% Confidence Intervals for Treatment Means
  
 --------------------------------------------------------------------------------
                                          Treatment          Lower          Upper
            Treatment      Treatment         Sample     Confidence     Confidence
     I           Mean             SD           Size          Limit          Limit
 --------------------------------------------------------------------------------
     1        5.34000        1.11714              5        4.43930        6.24070
     2        7.72000        1.10318              5        6.81930        8.62070
     3        8.56000        1.37768              5        7.65930        9.46070
     4        5.50000        0.97980              5        4.59930        6.40070
  
  
             95% Confidence Intervals for the Difference of Treatment Means
  
 ----------------------------------------------------------------------
             Difference of          Lower          Upper         Fisher
                 Treatment     Confidence     Confidence            LSD
     I    J          Means          Limit          Limit        P-Value
 ----------------------------------------------------------------------
     1    2       -2.38000       -3.65378       -1.10622        0.00489
     1    3       -3.22000       -4.49378       -1.94622        0.00043
     1    4       -0.16000       -1.43378        1.11378        0.82919
     2    3       -0.84000       -2.11378        0.43378        0.26651
     2    4        2.22000        0.94622        3.49378        0.00775
     3    4        3.06000        1.78622        4.33378        0.00069
  
  
             95% Bonferroni Intervals for the Difference of Treatment Means
  
 Bonferroni Adjusted Alpha = 0.0083
 ----------------------------------------------------------------------
     I    J  Difference of          Lower          Upper
                 Treatment     Confidence     Confidence
                     Means          Limit          Limit        P-Value
 ----------------------------------------------------------------------
     1    2       -2.38000       -4.33021       -0.42979        0.00489
     1    3       -3.22000       -5.17021       -1.26979        0.00043
     1    4       -0.16000       -2.11021        1.79021        0.82919
     2    3       -0.84000       -2.79021        1.11021        0.26651
     2    4        2.22000        0.26979        4.17021        0.00775
     3    4        3.06000        1.10979        5.01021        0.00069
  
  
             Tukey-Kramer Honest Significant Difference (HSD)
  
 Multiple Comparisons: alpha = 0.05
 ----------------------------------------------------------------------
              |Difference|                         Lower          Upper
              of Treatment       Standard     Confidence     Confidence
     I    J          Means          Error          Limit          Limit
 ----------------------------------------------------------------------
     1    2        2.38000        0.51590        0.28038        4.47962
     1    3        3.22000        0.51590        1.12038        5.31962
     1    4        0.16000        0.51590       -1.93962        2.25962
     2    3        0.84000        0.51590       -1.25962        2.93962
     2    4        2.22000        0.51590        0.12038        4.31962
     3    4        3.06000        0.51590        0.96038        5.15962
    

 
. Random Effects Case
.
. Step 1:   Create the data
.
read x y
 1     74
 1     76
 1     75
 2     68
 2     71
 2     72
 3     75
 3     77
 3     77
 4     72
 4     74
 4     73
 5     79
 5     81
 5     79
end of data
.
set let cross tabulate collapse
let ymean = cross tabulate mean y x
let ysd   = cross tabulate sd   y x
let yn    = cross tabulate size x
.
. Step 2:   Run random effects ANOVA
.
echo on
capture ONEWAYAN.OUT
set write decimals 6
one way random effects anova y x
one way summary random effects anova ymean ysd yn
end of capture
echo off
    

The following output is generated in ONEWAYA2.OUT

       ****************************************
       **  one way random effects anova y x  **
       ****************************************
  
  
             One-Way Random Effects Analysis of Variance: Raw Data
                          H0: Variance of Effects = 0
  
 Group-ID Variable:    Y
  
 Summary Statistics:
 Total Number of Observations                         15
 Number of Levels:                                     5
 Grand Mean:                                   74.866667
 Grand Standard Deviation:                      3.440653
  
  
  
             ANOVA Table: Fixed Effects
  
 --------------------------------------------------------------------------------------------------
               Source   DF Sum of Squares    Mean Square    F Statistic   F CDF (%)     P-Value Sig
 --------------------------------------------------------------------------------------------------
    Total (Corrected)   14     165.733333
 --------------------------------------------------------------------------------------------------
  Treatment (Between)    4     147.733333      36.933333        20.5185     99.9918    0.000082  **
    Residual (Within)   10      18.000000       1.800000
  
  
             Variance Components Table: Random Effects
  
 ----------------------------------------------------------------------------------------------------------------
                                              Lower          Upper       Intraclass          Lower          Upper
                            Variance     Confidence     Confidence      Correlation     Confidence     Confidence
               Source      Component          Limit          Limit  Coefficient (%)          Limit          Limit
 ----------------------------------------------------------------------------------------------------------------
  Treatment (Between)      11.711111       3.456828     101.096603            86.68      54.490128      98.364796
    Residual (Within)       1.800000       0.878770       5.543625            13.32
  
  
       *********************************************************
       **  one way summary random effects anova ymean ysd yn  **
       *********************************************************
  
  
             One-Way Random Effects Analysis of Variance: Summary Data
                            H0: Variance of Effects = 0
  
 Means Variable:       YMEAN
 SD Variable:          YSD
 Sample Size Variable: YN
  
 Summary Statistics:
 Total Number of Observations                         15
 Number of Levels:                                     5
 Grand Mean:                                   74.866667
  
  
  
             ANOVA Table: Fixed Effects
  
 --------------------------------------------------------------------------------------------------
               Source   DF Sum of Squares    Mean Square    F Statistic   F CDF (%)     P-Value Sig
 --------------------------------------------------------------------------------------------------
    Total (Corrected)   14     165.733333
 --------------------------------------------------------------------------------------------------
  Treatment (Between)    4     147.733333      36.933333        20.5185     99.9918    0.000082  **
    Residual (Within)   10      18.000000       1.800000
  
  
             Variance Components Table: Random Effects
  
 ----------------------------------------------------------------------------------------------------------------
                                              Lower          Upper       Intraclass          Lower          Upper
                            Variance     Confidence     Confidence      Correlation     Confidence     Confidence
               Source      Component          Limit          Limit  Coefficient (%)          Limit          Limit
 ----------------------------------------------------------------------------------------------------------------
  Treatment (Between)      11.711111       3.456828     101.096603            86.68      54.490128      98.364796
    Residual (Within)       1.800000       0.878770       5.543625            13.32