ENVIRONMENTAL SAMPLING AND ANALYSIS
EXPERIMENTAL DESIGN AND DATA ANALYSIS FOR BIOLOGISTS
WORKSHEET 3: Complex ANOVA 1: Nested and factorial designs
QUESTION 1: Here is the example of a nested design from lectures. Andrew & Underwood (1993) studied the effects of sea urchin grazing on a shallow subtidal reef in New South Wales, Australia. They set up four urchin density treatments (0% original, 33% original, 66% original, 100% original), with four patches (3 to 4 m2) of reef for each treatment and five quadrats from each patch. The response variable was % cover of filamentous algae in each quadrat. This is a nested design with treatment (fixed factor), patch nested within treatment (random factor) and quadrats as the residual. The data are in ANDREW. 1. What linear model is appropriate? 2. What are the main hypotheses to be tested here? H0 Effect 1: H0 Effect 2: _________________________________________________________________ _________________________________________________________________
3. What are the assumptions for this analysis? How can you examine them? There were large differences in within-cell variances. These data are %, although an arcsin√ had no effect in improving variance homogeneity, nor did a log transformation. So we will analyse untransformed data. 4. Fit the model and complete the table below Source of variation Treatment Patch(Treatment) Residual 5. Check that you have treated Patch as a random factor in completing this table. 6. Calculate the variance components (see Q&K Table 9.5) for treatments, patches and replicate quadrats (residual) and add them to ANOVA table. Where is the major variation in % cover of algae? SS df MS F P Variance component
Worksheet 3 Complex ANOVA 1: nested and factorial designs
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7. How might this information influence the way they might have designed future sampling?
QUESTION 2: In an unusually detailed preparation for an Environmental Effects Statement for a proposed discharge of dairy wastes into the Curdies River, in western Victoria, a team of stream ecologists wanted to describe the basic patterns of variation in a stream invertebrate thought to be sensitive to nutrient enrichment. As an indicator species, they focused on a small flatworm, Dugesia, and started by sampling populations of this worm at a range of scales. They sampled in two seasons, representing different flow regimes of the river - winter and summer. Within each season, they sampled three randomly chosen (well, haphazardly, because sites are nearly always chosen to be close to road access) sites. A total of six sites in all were visited, 3 in each season. At each site, they sampled six stones, and counted the number of flatworms on each stone. The data are in the file CURDIES, which contains two predictor variables (factors), SEASON$ (Winter & Summer) and SITE (1-3 in winter, 4-6 in summer), and the dependent variable, DUGESIA, containing the number of flatworms on a particular stone. After looking at the data, they field team decided that a transformation was necessary, and opted for the 4th-root transformation. The transformed data are in the variable S4DUGES. 1. What linear model is appropriate? 2. What are the main hypotheses to be tested here? H0 Effect 1: H0 Effect 2: _________________________________________________________________ _________________________________________________________________
3. What are the assumptions for this analysis? How can you examine them? 4. Fit the model and examine the residuals for both untransformed and transformed variables. If you agree with the transformation, go ahead and run the analysis using the variable S4DUGES. Complete the table below Source of variation SS df MS F P Variance component
Season Sites(Season) Residual 5. Check that you have treated Sites as a random factor in completing this table. 6. Calculate the variance components (see Q&K Table 9.5) for seasons, sites and replicate stones (residual) and add them to ANOVA table. Where is the major variation in numbers of flatworms? Between seasons? Sites? Stones?
Worksheet 3 Complex ANOVA 1: nested and factorial designs
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7. How might this information influence the way you designed future sampling?
QUESTION 3 Again, here is the factorial design we analysed in lectures. Quinn (1988) examined the effects of season (spring & summer) and adult density (8, 15, 30 & 45 animals per 225cm-2 enclosure) on the production of egg masses by intertidal pulmonate limpets (Siphonaria diemenensis). There were three replicate enclosures per treatment combination and the response variable was the number of egg masses per limpet in each enclosure. The data are in the file SIPHO. 1. What is the appropriate linear model for this design? 2. What null hypotheses are being tested by this 2 factor ANOVA? H0 main effect 1: H0 main effect 2: H0 interaction: 3. Fit the ANOVA model and save the residuals. Plot the residuals against the group means (predicted values or estimates): • Any evidence of wedge-shaped pattern (skewness) or outliers? 4. If the assumptions are OK, complete the table below from the results of fitting the ANOVA model: Source Density Linear Quadratic Season Density*Season Residual 5. Draw a bar or line graph with mean no. egg masses on the y-axis, Density on the x-axis and the 2 seasons as different lines (or different bars) – this is sometimes termed an interaction plot. What are your conclusions from the ANOVA model and the graph? 6. As part of the original design, we expected a strong effect of density and wanted to also test whether the effect represented a linear or quadratic trend in egg mass production with changing density. Test for linear and quadratic trends with density by partitioning the density SS into linear and quadratic components and complete the ANOVA table – what are your conclusions? NOTE: remember to use the MS Residual from original ANOVA to test these trends. SS df MS F P
Worksheet 3 Complex ANOVA 1: nested and factorial designs
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QUESTION 4 An ecologist studying a rocky shore at Phillip Island, in southeastern Australia, was interested in how clumps of intertidal mussels are maintained. In particular, he wanted to know how densities of adult mussels affected recruitment of young individuals from the plankton. As with most marine invertebrates, recruitment is highly patchy in time, so he expected to find seasonal variation, and the interaction between season and density - whether effects of adult mussel density vary across seasons was the aspect of most interest. The data were collected from four seasons, and with two densities of adult mussels. The experiment consisted of clumps of adult mussels attached to the rocks. These clumps were then brought back to the laboratory, and the number of baby mussels recorded. There were 3-6 replicate clumps for each density and season combination. The data are in the file RECRUIT. The independent variables are PERIOD, coded 1, 2, 3, and 4, for Spring, Summer, Autumn, and Winter, and DENSITY, with values 1 and 2 for low and high densities of adults. The number of recruits is contained in the variable RECRUITS. After initial inspection of box plots, it was decided to use a square-root transformation on the recruitment data. The transformed variable is labelled SRECS. 1. What linear model is appropriate for this data set and analysis? 2. Fit this model using untransformed data and save the residuals. Plot these residuals (residuals against predicted values) and check that our decision to transform the data was sensible. If you agree, proceed with the analysis using SRECS. 3. Fit the model using transformed data and complete the table below. Was there a significant interaction between DENSITY and PERIOD? How would you interpret this interaction? Source of variation Period Density Period * Density Effect of density in summer Effect of density in autumn Effect of density in winter Effect of density in spring Residual SS df MS F P
4. Let’s explore the interaction a bit further. Do an “interaction plot” of cell means with PERIOD on the x-axis, SRECS on the y-axis and different bars or lines for each DENSITY. What are your conclusions? 5. Now let’s use simple main effects to examine the effect of density separately at each season, OR the effect of season separately at each mussel density. Which approach is of more interest in this case? Add the results of the appropriate tests to the ANOVA table above – again, what are your conclusions? NOTE: make sure you use the MS Residual from original ANOVA to test these simple main effects!
Worksheet 3 Complex ANOVA 1: nested and factorial designs
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QUESTION 5 An experiment was set up to test the hypothesis that healthy spruce seedlings break bud sooner than diseased spruce seedlings. There were 2 factors: pH (3 levels: 3, 5.5, 7) and HEALTH (2 levels: healthy, diseased). The dependent variable was the average (from 5 buds) bud emergence rating (BRATING) on each seedling. The sample size varied for each combination of pH and health, ranging from 7 to 23 seedlings (why aren’t the 5 buds from each tree true replicates?). With two factors, this experiment should be analysed with a 2 factor (2 x 3) ANOVA. The data are in the files SPRUCE. There is an additional variable in the file called GROUP, which represents each pH and health combination. 1. Check the assumptions of the analysis. Do a boxplot of BRATING for each group (simply boxplot brating by group). Any evidence of skewness or outliers? We will double check the assumptions when we examine residuals below. 2. What is the appropriate linear model for this design? 3. What null hypotheses are being tested by this 2 factor ANOVA? H0 main effect 1: H0 main effect 2: H0 interaction: 4. Fit the ANOVA model and save the residuals. Plot the residuals against the group means (predicted values or estimates): • Any evidence of wedge-shaped pattern (skewness) or outliers? 5. If the assumptions are OK, complete the table below: Source of variation pH Health pH * Health Residual SS df MS F P
6. Draw a bar or line graph with mean brating on the y-axis, pH on the x-axis and the 2 disease states (healthy, diseased) as different lines (or different bars). 7. What conclusions would you draw from the graph and the ANOVA?
Worksheet 3 Complex ANOVA 1: nested and factorial designs
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