Homework 6 (NESTEDREPEATED MEASURES and ANCOVA) answer key page 1

Worksheet 6: a brief exercise on NESTED ANOVA and Analysis of Covariates – KEY

1)NESTED (Variance components)

a)The square root transform is the best (try Log if you want to). A square root transform has a similar effect to that of Log transforms. Sometimes Log transforms over correct – if this occurs try a square root transform.

c)They are random effects because we do not care (at all) about the actual spatial coordinates – instead we care about what spatial scales they REPRESENT.

e)There are 5 terms in the table. Four are obvious but note also that the Residual term actually models the plot scale.. Most of the variance, 72%, (see last column) is at the smallest spatial scale (plot) 16.5% is at the site scale and 11.5% is at the regional scale.

f)COME UP WITH SOME

2)Repeated Measures

3) ANCOVA –

i. Full model

The interaction between TREATMENT and DISTANCE is not significant, hence the slopes are homogeneous.

ii. Reduced Model

Now you can see that both TREATMENT and DISTANCE are significant.

  1. The presentation would depend on your hypothesis. Here are some possibilities

1)HA1: There is a positive effect of distance and watering on seedlings

Here you would want to show the following two graphs along with the reduced ANOVA table. First the bar graph of the effect of treatment on seedling number.

You need to use the least square means to get the correct values (adjusted for the effect of the covariate). In the output from the reduced model the Least Square means will be in a table under TREATMENT. Right click in the middle of the table and Make a new data table. Change the name ‘Level’ to TREATMENT. Then make a bar graph in GRAPH, CHART.

The second graph is just the relationship between distance and seedlings. Use the original data and create a scatterplot. Use the X and Y axis tabs to label the axis and use a linear smoother. Use the GRAPH BUILDER.

2)HA2: After accounting for distance there is a linear relationship between seedling number and watering treatment.

Here we really don’t care about distance – we just want to remove that source of variance from the analysis. First you want to present the reduced model ANOVA table. This indicates that both Treatment and Distance are important in determining the number of seedlings. Now you want to formally test for a linear trend. Here you are going to use the CONTRAST testing feature found under TREATMENT (in the output page from the reduced model). Click on LS MEANS CONTRAST. Then use the following coefficients (the is a polynomial 1st order) . Run the analysis

There is support for the linear trend.