Question

1	1	111.5
1	2	97.7
1	3	126.1
2	1	94.4
2	2	70.5
2	3	93.1
3	1	73.9
3	2	56.2
3	3	84.6

In many agricultural and biological experiments, one may use a two‑way model with only one observation per cell. When one of the factors is related to the grouping of experimental units into more uniform groups, the design may be called a randomized complete block design (RCBD). The analysis is similar to a two‑way analysis of variance (question B) except that the model does not include an interaction term.

            The specific leaf areas (area per unit mass) of three types of citrus each treated with one of three levels of shading are stored in Table C. The first column contains the code for the shading treatment, the second column contains the code for the citrus species, and the third column contains the specific leaf area. Assume that there is no interaction between citrus species and shading. Carry out a two‑way analysis of this data.

            The shading treatment and citrus species are coded as follows:

            Treatment        Code                Species                        Code

            Full sun           1                      Shamouti orange         1

            Half shade       2                      Marsh grapefruit          2

            Full shade        3                      Clementine mandarin 3

nCopy the treatment code, the species code, and the specific leaf area into the EXCEL worksheet, label the columns and look at the data.

                                                                                                                                       {Example 1}

nPerform a two‑way (without interaction) analysis of this data and answer the following questions. Use a 5% significance level.

Source of variation	Degrees of freedom		Sum of squares		Mean square		F		P
Shading treatment	2
Citrus species	2
Error	4

24. Should the hypothesis that shading treatment has no effect on specific leaf area be rejected (1) or not (0)?

25. Should the hypothesis that citrus species do not differ in specific leaf area be rejected (1) or not (0)?

26. What is the estimate of the average (pooled) variance in this experiment (i.e. Error mean square)?

27. What are the error degrees of freedom for the pooled variance?

{Example 26}

            Recall that the confidence interval for a difference between two means is based on a calculation of the margin of error of the estimated difference. With a common variance (Error MS) and the same number of observations in all shading treatments, the margin of error of an estimated difference will be the same whether we calculate it for treatments 1 and 2, 1 and 3, or 2 and 3. This margin of error of the difference between two means is sometimes referred as the least significant difference (LSD).

nCalculate the LSD for comparing shading treatments in this experiment.

LSD = critical tvalue ´standard error of difference.

Use the critical t value with 4 degrees of freedom is t _0.025,4= 2.776.

n is the number of times of times each treatment was tested (in this case n = 3 for the 3 species).

28. What is the least significant difference (a = 0.05) for comparing shading treatments in this experiment?

Answer 1

Using Excel toolpak:

Two-way Anova:

(24): Shading treatment p-value is less than 0.05, so we reject the null hypothesis.

(25): Citrus Species p-value is less than 0.05, so we reject the null hypothesis.

(26):

pooled variance = MSE/degree freedom of the error

pooled variance = 17.638/4

pooled variance = 4.40

(27): Error degrees of freedom for the pooled variance is 4.

(28): Using SPSS:

All treatments are significant each other because p value is less than 0.05.