Question

The algae dataset (last page) uses repeated measures: chlorophyll concentrations are measured at three depths in each of 4 lakes. Fit two models to test for an effect of depth on chlorophyll concentration: one with only depth as a predictor, and one with both depth as the predictor and lake as the “subject”. Produce ANOVA tables for both models. Create a graph showing how chlorophyll concentration changes with depth. Describe each result, then compare them and explain any differences.

Before you do any of this, make sure your depth variable is being treated as a categorical variable, with a logical order for the values.

algae

Lake	surface	1 m	3 m
1	425	130	56
2	500	215	115
3	100	30	10
4	325	100	28

Answer 1

From the above box-plot, the chlorophyll concentrations are changed with the change of depth. The concentration is high which is collected from the surface and decreased with increasing the depth.

a) The ANOVA table considering only the depth as the subject

Source	DF	SS	MS	F	P
Depth	2	178189	89094	7	0.015
Error	9	114569	12730
Total	11	292757

Comment: The estimated p-value for the F-test statistic is 0.015. Hence, we can conclude that at least a depth level has significant mean chlorophyll concentrations at the 0.05 significance level.

b)

Two-way ANOVA: chlorophyll versus Depth, Lake

Source	DF	SS	MS	F	P
Depth	2	178189	89094.3	17.63	0.003
Lake	3	84247	28082.3	5.56	0.036
Error	6	30322	5053.6
Total	11	292757

Comment: From the Above table, the estimated p-value for the F-test statistic for the subject depth is 0.003. Hence, we can conclude that at least a depth level has significant mean chlorophyll concentrations at the 0.05 significance level. Similarly, the estimated p-value for the F-test statistic for the subject Lake is 0.036. Hence, we can conclude that at least a Lake has significant mean chlorophyll concentrations at the 0.05 significance level.