Question

Iron-deficiency anemia is the most common form of malnutrition in developing countries, affecting about 50% of children and women and 25% of men. Iron pots for cooking foods had traditionally been used in many of these countries, but they have been largely replaced by aluminum pots, which are cheaper and lighter. Some research has suggested that food cooked in iron pots will contain more iron than food cooked in other types of pots. One study designed to investigate this issue compared the iron content of some Ethiopian foods cooked in aluminum, clay, and iron pots. Foods considered were yesiga wet', beef cut into small pieces and prepared with several Ethiopian spices; shiro wet', a legume-based mixture of chickpea flour and Ethiopian spiced pepper; and ye-atkilt allych'a, a lightly spiced vegetable casserole. Four samples of each food were cooked in each type of pot. The iron in the food is measured in milligrams of iron per 100 grams of cooked food. The data are shown in the table below.

		Iron Content (mg/100 g) of Food Cooked in Different Pots
Type of pot		Meat					Legumes					Vegetables
Aluminum		1.77	2.36	1.96	2.14		2.40	2.17	2.41	2.34		1.03	1.53	1.07	1.30
Clay		2.27	1.28	2.48	2.68		2.41	2.43	2.57	2.48		1.55	0.79	1.68	1.82
Iron		5.27	5.17	4.06	4.22		3.69	3.43	3.84	3.72		2.45	2.99	2.80	2.92

(a) Make a table giving the sample size, mean, and standard deviation for each type of pot. Is it reasonable to pool the variances? Although the standard deviations vary more than we would like, this is partially due to the small sample sizes, and we will proceed with the analysis of variance.

(b) Plot the means. Give a short summary of how the iron content of foods depends upon the cooking pot.

(c) Run the analysis of variance. Give the ANOVA table, the F statistics with degrees of freedom and P-values, and your conclusions regarding the hypotheses about main effects and interactions.

Answer 1

(a)

Test for Equal Variances: iron in the food versus Type of pots

95% Bonferroni confidence intervals for standard deviations

Type of pots N Lower StDev Upper
Aluminum 12 0.344609 0.522274 1.01262
Clay 12 0.396777 0.601337 1.16592
Iron 12 0.583477 0.884291 1.71453

Bartlett's Test (Normal Distribution)
Test statistic = 3.28, p-value = 0.194>0.05

Levene's Test (Any Continuous Distribution)
Test statistic = 0.92, p-value = 0.407>0.05

From the above two tests, p-value>0.05, hence it is reasonable to pool the variances.

(b)

From the above diagram, it is observed that iron content of foods depends upon the cooking pot. If cooking pot is iron, then the average iron content of foods is largest than other two pots. Whereas the average iron content of foods is more or less same for Aluminum and Clay pots. However the average  iron content of foods for Aluminum pot is less than the average  iron content of foods for Clay pot.

(c)

Two-way ANOVA: iron in the food versus Type of pots, Food

Source DF SS MS F P
Type of pots 2 24.8940 12.4470 92.26 0.000
Food 2 9.2969 4.6484 34.46 0.000
Interaction 4 2.6404 0.6601 4.89 0.004
Error 27 3.6425 0.1349
Total 35 40.4738

From the ANOVA table, we see that p-value for interaction effect=0.004<0.05, so iteraction effect is significantly present. Hence it is meaningless to test the main effects separately.