THE COMPLETELY RANDOMIZED DESIGN (CRD)

LAB # 1

DEFINITIONAchieved when the samples of

experimental units for each treatment are random and independent of each other

Design is used to compare the treatment means:

0 1 2: ... kH :aH At least two of the treatment means differ

•The hypotheses are tested by comparing the differences between the treatment means.

•Test statistic is calculated using measures of variability within treatment groups and measures of variability between treatment groups

STEPS FOR CONDUCTING AN ANALYSIS OF VARIANCE (ANOVA) FOR A COMPLETELY RANDOMIZED DESIGN:

•1 -Assure randomness of design, and independence, randomness of samples

•2 -Check normality, equal variance assumptions

•3 -Create ANOVA summary table

•4 -Conduct multiple comparisons for pairs of means as necessary/desired

ASSUMPTIONS

1 -Normality :

You can check on normality using

1 -plot

2 -Kolmogorve test

2 -Constant variance:

You can check on homogeneity of variances using

1 -Plot2 -leven’s test.

ONE WAY ANOVA

ANOVA Summary Table for a Completely Randomized Design

Source df SS MS F

Treatments 1k SST 1

SSTMST

k

MST

MSE

Error n k SSE SSE

MSEn k

Total 1n SS Total

MULTIPLE COMPARISONS OF MEANS

•A significant F-test in an ANOVA tells you that the treatment means as a group are statistically different.

•Does not tell you which pairs of means differ statistically from each other

•With k treatment means, there are c different pairs of means that can be compared, with c calculated as

1

2

k kc

MULTIPLE COMPARISONS OF MEANS

Guidelines for Selecting a Multiple Comparisons Method in ANOVA

Method Treatment Sample Sizes Types of Comparisons Tukey Equal Pairwise Bonferroni Equal or Unequal Pairwise Scheffe Equal or Unequal General Contrasts

EXAMPLE 1

A manufacturer of television sets is interested in the effect on tube conductivity of four different types of coating for color picture tubes. The

following conductivity data are obtained.

CoatingConductivity

1146150141143

2143137149152

3127132136134

4129132127129

SOLUTION

Enter data in spss as follows:

ANALYSIS

Test of Homogeneity of Variances

conductiivity

Levene Statisticdf1df2Sig.

2.370312.122

Tests of Normality

Kolmogorov-SmirnovaShapiro-Wilk

StatisticdfSig.StatisticdfSig.

conductiivity.13316.200*.92816.230

a. Lilliefors Significance Correction

*. This is a lower bound of the true significance.

ONE WAY ANOVA

ANOVA

conductiivity

Sum of SquaresdfMean SquareFSig.

Between Groups844.6883281.56214.302.000

Within Groups236.2501219.688

Total1080.93815

Multiple Comparisons

Dependent Variable:conductiivity

(I) coating(J) coating

Mean Difference (I-

J)Std. ErrorSig.

95% Confidence Interval

Lower BoundUpper Bound

Tukey HSD12-.250-3.1371.000-9.56-9.06

312.750*3.137.0073.4422.06

415.750*3.137.0016.4425.06

21.2503.1371.000-9.06-9.56

313.000*3.137.0063.6922.31

416.000*3.137.0016.6925.31

31-12.750*3.137.007-22.06--3.44-

2-13.000*3.137.006-22.31--3.69-

43.0003.137.776-6.31-12.31

41-15.750*3.137.001-25.06--6.44-

2-16.000*3.137.001-25.31--6.69-

3-3.000-3.137.776-12.31-6.31

LSD12-.250-3.137.938-7.09-6.59

312.750*3.137.0025.9119.59

415.750*3.137.0008.9122.59

21.2503.137.938-6.59-7.09

313.000*3.137.0016.1619.84

416.000*3.137.0009.1622.84

31-12.750*3.137.002-19.59--5.91-

2-13.000*3.137.001-19.84--6.16-

43.0003.137.358-3.84-9.84

41-15.750*3.137.000-22.59--8.91-

2-16.000*3.137.000-22.84--9.16-

3-3.000-3.137.358-9.84-3.84

Thanks for all