6.9: Try It!

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Exercise \(\PageIndex{1}\)

Three teaching methods were to be compared to teach computer science in high schools. Nine different schools were chosen randomly and each teaching method was assigned to 3 randomly chosen schools so that each school implemented only one teaching method. The response that was used to compare the 3 teaching methods was the average score for each high school.

Show data Lesson6_1ex1

    data Lesson6_ex1;
    input mtd school score semester $;
    datalines;
    1 1 68.11 Fall
    1 1 68.11 Fall
    1 1 68.21 Fall
    1 1 78.11 Spring
    1 1 78.11 Spring
    1 1 78.19 Spring
    1 2 59.21 Fall
    1 2 59.13 Fall
    1 2 59.11 Fall
    1 2 70.18 Spring
    1 2 70.62 Spring
    1 2 69.11 Spring
    1 3 64.11 Fall
    1 3 63.11 Fall
    1 3 63.24 Fall
    1 3 63.21 Spring
    1 3 64.11 Spring
    1 3 63.11 Spring
    2 1 84.11 Fall
    2 1 85.21 Fall
    2 1 85.15 Fall
    2 1 85.11 Spring
    2 1 83.11 Spring
    2 1 89.21 Spring
    2 2 93.11 Fall
    2 2 95.21 Fall
    2 2 96.11 Fall
    2 2 95.11 Spring
    2 2 97.27 Spring
    2 2 94.11 Spring
    2 3 90.11 Fall
    2 3 88.19 Fall
    2 3 89.21 Fall
    2 3 90.11 Spring
    2 3 90.11 Spring
    2 3 92.21 Spring
    3 1 74.2 Fall
    3 1 78.14 Fall
    3 1 74.12 Fall
    3 1 87.1 Spring
    3 1 88.2 Spring
    3 1 85.1 Spring
    3 2 74.1 Fall
    3 2 73.14 Fall
    3 2 76.21 Fall
    3 2 72.14 Spring
    3 2 76.21 Spring
    3 2 75.1 Spring
    3 3 80.12 Fall
    3 3 79.27 Fall
    3 3 81.15 Fall
    3 3 85.23 Spring
    3 3 86.14 Spring
    3 3 87.19 Spring
    ;

Using the information about the teaching method, school, and score only, the school administrators conducted a statistical analysis to determine if the teaching method had a significant impact on student scores. Perform a statistical analysis to confirm their conclusion.
If possible, perform any other additional statistical analyses.

Show Solution in SAS

1. To confirm their conclusion, a model with only the two factors, teaching method and school was used, with school nested within the teaching method.

Input:

data Lesson6_ex1;
    input mtd school score semester $;
    datalines;
    1 1 68.11 Fall
    1 1 68.11 Fall
    1 1 68.21 Fall
    1 1 78.11 Spring
    1 1 78.11 Spring
    1 1 78.19 Spring
    1 2 59.21 Fall
    1 2 59.13 Fall
    1 2 59.11 Fall
    1 2 70.18 Spring
    1 2 70.62 Spring
    1 2 69.11 Spring
    1 3 64.11 Fall
    1 3 63.11 Fall
    1 3 63.24 Fall
    1 3 63.21 Spring
    1 3 64.11 Spring
    1 3 63.11 Spring
    2 1 84.11 Fall
    2 1 85.21 Fall
    2 1 85.15 Fall
    2 1 85.11 Spring
    2 1 83.11 Spring
    2 1 89.21 Spring
    2 2 93.11 Fall
    2 2 95.21 Fall
    2 2 96.11 Fall
    2 2 95.11 Spring
    2 2 97.27 Spring
    2 2 94.11 Spring
    2 3 90.11 Fall
    2 3 88.19 Fall
    2 3 89.21 Fall
    2 3 90.11 Spring
    2 3 90.11 Spring
    2 3 92.21 Spring
    3 1 74.2 Fall
    3 1 78.14 Fall
    3 1 74.12 Fall
    3 1 87.1 Spring
    3 1 88.2 Spring
    3 1 85.1 Spring
    3 2 74.1 Fall
    3 2 73.14 Fall
    3 2 76.21 Fall
    3 2 72.14 Spring
    3 2 76.21 Spring
    3 2 75.1 Spring
    3 3 80.12 Fall
    3 3 79.27 Fall
    3 3 81.15 Fall
    3 3 85.23 Spring
    3 3 86.14 Spring
    3 3 87.19 Spring
    ;
proc mixed data=lesson6_ex1 method=type3;
class mtd school;
model score = mtd;
random school(mtd);
store results1;
run;

proc plm restore=results1;
lsmeans mtd / adjust=tukey plot=meanplot cl lines;
run;

Partial outputs:


Type 3 Analysis of Variance
Source	DF	Sum of Squares	Mean Square	Expected Mean Square	Error Term	Error DF	F Value	Pr > F
mtd	2	4811.400959	2405.700480	Var(Residual) + 6 Var(school(mtd)) + Q(mtd)	MS(school(mtd))	6	16.50	0.0036
school(mtd)	6	875.059744	145.843291	Var(Residual) + 6 Var(school(mtd))	MS(Residual)	45	10.13	<.0001
Residual	45	647.972350	14.399386	Var(Residual)	.	.	.	.

The \(p\)-value of .0036 indicates that the scores vary significantly among the 3 teaching methods and confirms the school administrators’ conclusion. As the teaching method was significant, the Tukey procedure was conducted to determine the significantly different pairs among the 3 teaching methods. The results of the Tukey procedure shown below indicate that the mean scores of teaching methods 2 and 3 are not statistically significant and that the teaching method 1 mean score is statistically lower than the mean scores of the other two.

SAS-generated graph of score Tukey groupings for LS-means of mtd. Mtd values of 2 and 3 are covered by a single blue bar, while the mtd value of 1 is covered by a single red bar. — Figure \(\PageIndex{a1}\): LS-means of mtd score Tukey grouping.

Plot of all pairwise score least-squares means differences for mtd with Tukey adjustment at significance level 0.05. — Figure \(\PageIndex{a2}\): Diffogram of score comparisons for mtd with Tukey adjustment.

2. Using the additional code shown below, an ANOVA was conducted including semester also as a possible fixed effect.

proc mixed data=lesson6_ex1 method=type3;
class mtd school semester ;
model score = mtd semester mtd*semester;
random school(mtd) semester*school(mtd);
store results2;
run;

proc plm restore= results2;
lsmeans mtd semester / adjust=tukey plot=meanplot cl lines;
run;

The \(p\)-values indicate that both these main effects are statistically significant, but not their interaction. The Tukey procedure indicates that the significances of paired comparisons for the teaching method remain the same. Between the two semesters, the scores are statistically higher in the spring compared to the fall.

Note

The output writes semester*school(mtd) as school*semester(mtd), probably due to arranging effects in alphabetical order.


semester Least Squares Means
semester	Estimate	Standard Error	DF	t Value	Pr > \|t\|	Alpha	Lower	Upper
Fall	76.6370	1.8265	6	41.96	<.0001	0.05	72.1677	81.1063
Spring	81.2411	1.8265	6	44.48	<.0001	0.05	76.7718	85.7104

Show Solution in Minitab

1. Choose Stat -> ANOVA -> General Linear Model

Minitab General Linear Model pop-up window, with "score" in the Responses window and "mtd-school" in the Factors window. — Figure \(\PageIndex{b1}\): Minitab General Linear Model pop-up window.

Then, click Random/Nest:

Minitab General Linear Model window for Random/Nest, with "mtd" entered next to the factor of "school" in the Nesting table, mtd set as a fixed factor, and school set as a random factor. — Figure \(\PageIndex{b2}\): General Linear Model: Random/Nest pop-up window.

Output:

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
mtd	2	4811.4	2405.70	16.50	0.004
school(mtd)	6	875.1	145.84	10.13	0.000
Error	45	648.0	14.40
Total	53	6334.4

Conclusion

The \(p\)-value of .004 indicates that mtd is statistically significant, which implies that the mean score from all 3 teaching methods is not the same, thus confirming the school administrators’ claim. Note that in the Minitab General Linear Model, the Tukey procedure or any other paired comparisons are not available.

2. Choose Stat -> ANOVA -> General Linear Model

Minitab General Linear Model pop-up window with "score" in the Responses window and "mtd-school semester" in the Factors window. — Figure \(\PageIndex{b3}\): Minitab General Linear Model pop-up window.

Then click Random/Nest.

Minitab General Linear Model window for Random/Nest, with "mtd" entered next to "school" in the Nesting table, "mtd" and "semester" set as fixed factors, and "school" set as a random factor. — Figure \(\PageIndex{b4}\): General Linear Model: Random/Nest pop-up window.

Hit OK and then click Model

Minitab GLM: Model window, with "2" selected in the Interactions through order window. — Figure \(\PageIndex{b5}\): General Linear Model: Model pop-up window.

Select the effects mtd, semester, and school(mtd), and then click Add.

GLM Model window with the selected factors of "mtd", "school(mtd)", and "semester." — Figure \(\PageIndex{b6}\): General Linear Model: Model pop-up window, with selected effects.

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
mtd	2	4811.40	2405.70	16.50	0.004
semester	1	286.17	286.17	8.34	0.028
school(mtd)	6	875.06	145.84	4.25	0.051
mtd*semester	2	85.70	42.85	1.25	0.352
school(mtd)*semester	6	205.85	34.31	17.58	0.000
Error	36	70.25	1.95
Total	53	6334.43

Conclusion

The \(p\)-values indicate that both main effects, mtd and semester, are statistically significant, but not their interaction. Note that in the Minitab General Linear Model procedure, paired comparisons are not available.

Exercise \(\PageIndex{2}\)


Type 3 Analysis of Variance
Source	DF	Sum of Squares	Mean Square	Expected Mean Square	F Value	Pr > F
	2	4811.400959	2405.700480	Var(Residual) + 6 Var(A*B) + Q(A)	11.38	0.0224
	2	29.274959	14.637480	Var(Residual) + 6 Var(A*B) + 18 Var(B)	0.07	0.9342
	4	845.784785	211.446196	Var(Residual)+ 6 Var(A*B)	14.68	<.0001
Residual	45	647.972350	14.399386	Var(Residual)

Use the ANOVA table above to answer the following.

Name the fixed and random effects.
Complete the Source column of the ANOVA table above.
How many observations are included in this study?
How many replicates are there?
Write the model equation.
Write the hypotheses that can be tested with the expression for the appropriate \(F\)-statistic.

Show Solution

1. Name the fixed and random effects.

Fixed: A with 3 levels. In the EMS column, Q(A) reveals that A is fixed and the df indicates that it has 3 levels. Note that any factor that has a quadratic form associated with it is fixed and Q(A) is the quadratic form associated with A. This actually equals \(\sum_{i=1}^{3} \alpha_{i}^{2}\), where \(i = 1,2,3\) are the treatment effects; it is non-zero if the treatment means are significantly different.

Random: B is random as indicated by the presence of Var(B), The effect of factor B is studied by sampling 3 cases (see df value for B).

A*B is random as any effect involving a random factor is random.
The residual is also random as indicated by the presence of the Var(residual) in the EMS column.

2. Complete the Source column of the ANOVA table above.

Use the EMS column and start from the bottom row. The bottom-most has only var(*residual) and therefore the effect on the corresponding Source is residual. The next row up has var(A*B) in the additional term indicating that the corresponding source is A*B, etc.


Type 3 Analysis of Variance
Source	DF	Sum of Squares	Mean Square	Expected Mean Square	F Value	Pr > F
A	2	4811.400959	2405.700480	Var(Residual) + 6 Var(A*B) + Q(A)	11.38	0.0224
B	2	29.274959	14.637480	Var(Residual) + 6 Var(A*B) + 18 Var(B)	0.07	0.9342
A*B	4	845.784785	211.446196	Var(Residual)+ 6 Var(A*B)	14.68	<.0001
Residual	45	647.972350	14.399386	Var(Residual)	.	.

3. How many observations are included in this study?

\(N-1= 2+2+4+45=53\), so \(N=54\).

4. How many full replicates are there?

Let \(r\)=number of replicates. Then \(N\) = number of levels of A times number of levels of B times \(r\) = \(3 \times 3 \times r\). Therefore, \(9 \times r = 54\), which gives \(r=6\).

5. Write the model equation.

\(y_{ijk} = \mu + \alpha_{i} + \beta_{j} + (\alpha \beta)_{ij} + \epsilon_{ijk}\) where \(i, j = 1,2,3\) and \(k=1,2,\ldots,6\)

6. Write the hypotheses that can be tested with the \(F\)-statistic information.

	Effect A	Effect B	*Effect AB**
Hypotheses	\(H_{0}: \alpha_{i}=0 \text{ for all } i \text{ vs. } H_{a}: \alpha_{i} \neq 0\) for at least one \(i=1,2,3\) Note that \(\sum_{i=1}^{3} \alpha_{i}^{2}\) is the non-centrality parameter of the \(F\)-statistics if \(H_{a}\) is true.	\(H_{0}: \sigma_{\beta}^{2} = 0 \text{ vs. } H_{a}: \sigma_{\beta}^{2} > 0\)	\(H_{0}: \sigma_{\alpha \beta}^{2} = 0 \text{ vs. } H_{a}: \sigma_{\alpha \beta}^{2} > 0\)
\(F\) Statistic	\(\dfrac{2405.700480}{211.446196} = 11.377\) with 2 and 4 degrees of freedom	\(\dfrac{14.63480}{211.446196} = 0.0692\) with 2 and 4 degrees of freedom	\(\dfrac{211.446916}{14.399386} = 14.685\) with 4 and 45 degrees of freedom