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Here is a modified example from Quinn and Keough (2002). Day and Quinn (1989) described an experiment that examined how rock surface type affected the recruitment of barnacles to a rocky shore. The experiment had a single factor, surface type, with 4 treatments or levels: algal species 1 (ALG1), algal species 2 (ALG2), naturally bare surfaces (NB) and artificially scraped bare surfaces (S). There were 5 replicate plots for each surface type and the response (dependent) variable was the number of newly recruited barnacles on each plot after 4 weeks.
Open the day data file.
Note that as with independent t-tests, variables are in columns with levels of the categorical variable listed repeatedly. Day and Quinn (1989) were interested in whether substrate type influenced barnacle recruitment. This is a biological question. To address this question statistically, it is first necessary to re-express the question from a statistical perspective.
Using boxplots, examine the assumptions of normality and homogeneity of variance. Note that when sample sizes are small (as is the case with this data set), these ANOVA assumptions cannot reliably be checked using boxplots since boxplots require at least 5 replicates (and preferably more), from which to calculate the median and quartiles. As with regression analysis, it is the assumption of homogeneity of variances (and in particular, whether there is a relationship between the mean and variance) that is of most concern for ANOVA.
Here is a modified example from Quinn and Keough (2002). Medley & Clements (1998) studied the response of diatom communities to heavy metals, especially zinc, in streams in the Rocky Mountain region of Colorado, U.S.A.. As part of their study, they sampled a number of stations (between four and seven) on six streams known to be polluted by heavy metals. At each station, they recorded a range of physiochemical variables (pH, dissolved oxygen etc.), zinc concentration, and variables describing the diatom community (species richness, species diversity H and proportion of diatom cells that were the early-successional species, Achanthes minutissima). One of their analyses was to ignore streams and partition the 34 stations into four zinc-level categories: background (< 20 µg.l-1, 8 stations), low (21-50 µg.l-1, 8 stations), medium (51-200 µg.l-1, 9 stations), and high (> 200 µg.l-1, 9 stations) and test null hypotheses that there we no differences in diatom species diversity between zinc-level groups, using stations as replicates. We will also use these data to test the null hypotheses that there are no differences in diatom species diversity between streams, again using stations as replicates.
Open the medley data file.
Most statistical packages automatically order the levels of categorical variables alphabetically. Therefore, the levels of the ZINC categorical variable will automatically be ordered as (BACK, HIGH, LOW, MEDIUM). For some data sets the ordering of factors is not important. However, in the medley data set, it would make more sense if the factors were in the following order (BACK, LOW, MEDIUM, HIGH) as this would more correctly represent the relationships between the levels. Note that the ordering of a factor has no bearing on any analyses, it just makes the arrangement of data summaries within some graphs and tables more logical. It is therefore recommended that whenever a data set includes categorical variables, reorder the levels of these variables into a logical order.
If the assumptions seem reasonable, fit the linear model, check the residuals and if still there is no clear indication of problems, examine the ANOVA table.
Here is a modified example from Quinn and Keough (2002). Partridge and Farquhar (1981) set up an experiment to examine the effect of reproductive activity on longevity (response variable) of male fruitflies (Drosophila sp.). A total of 125 male fruitflies were individually caged and randomly assigned to one of five treatment groups. Two of the groups were used to to investigate the effects of the number of partners (potential matings) on male longevity, and thus differed in the number of female partners in the cages (8 vs 1). There were two corresponding control groups containing eight and one newly pregnant female partners (with which the male flies cannot mate), which served as controls for any potential effects of competition between males and females for food or space on male longevity. The final group had no partners, and was used as an overall control to determine the longevity of un-partnered male fruitflies.
Open the partridge data file.
Note, normally we might like to adjust the ordering of the levels of the categorical variable (GROUP), however, in this case, the alphabetical ordering also results in the most logical ordering of levels.
In addition to the global ANOVA in which the overall effect of the factor on male fruit fly longevity is examined, a number of other comparisons can be performed to identify differences between specific groups. As with the previous question, we could perform Tukey's post-hoc pairwise comparisons to examine the differences between each group. Technically, it is only statistically legal to perform n-1 pairwise comparisons, where n is the number of groups. This is because if each individual comparison excepts a 5% (&alpha=0.05) probability of falsely rejecting the H0, then the greater the number of tests performed the greater the risk eventually making a statistical error. Post-hoc tests protect against such an outcome by adjusting the &alpha values for each individual comparison down. Consequently, the power of each comparison is greatly reduced.
This particular study was designed with particular comparisons in mind, while other pairwise comparisons would have very little biological meaning or importance. For example, in the context of the project aims, comparing group 1 with group 2 would not yield anything meaningful. As we have five groups (df=4), we can do four planned comparisons.
Note that the Residual (within groups) term is common to each planned comparison as well as the original global ANOVA. Copy and paste the ANOVA table from the Rcmdr output window into Word and add an appropriate table caption .
Snodgrass et al. (2000) were interested in how the assemblages of larval amphibians varied between depression wetlands in South Carolina, USA, with different hydrological regimes. A secondary question was whether the presence of fish, which only occurred in wetlands with long hydroperiods, also affected the assemblages of larval amphibians. They sampled 22 wetlands in 1997 (they originally had 25 but three dried out early in the study) and recorded the species richness and total abundance of larval amphibians as well as the abundance of individual taxa. Wetlands were also classified into three hydroperiods: short (6 wetlands), medium (5) and long (11) - the latter being split into those with fish (5) and those without (6). The short and medium hydroperiod wetlands did not have fish.
The overall question of interest is whether species richness differed between the four groups of wetlands. However, there are also specific questions related separately to hydroperiod and fish. Is there a difference in species richness between long hydroperiod wetlands with fish and those without? Is there a difference between the hydroperiods for wetlands without fish? We can address these questions with a single factor fixed effects ANOVA and planned contrasts using species richness of larval amphibians as the response variable and hydroperiod/fish category as the predictor (grouping variable).
Open the snodgrass data file.
Reorder the factor levels of HYDROPERIOD into a more logical order (e.g. SHORT, MEDIUM, LONGNOFISH, LONGFISH)
A marine biologist was studying the effects of increased lead concentration ([Pb]) in the water on deformities in sperm cells of male flathead in Port Phillip Bay. The aquarium room at the Queenscliff Marine Station was used. There were two large tanks used for the experiment. In one tank (treatment tank), 50l of seawater was removed weekly and replaced with 50l of seawater with a [Pb] of 1 mg/l. In the other tank, the seawater was not altered (control tank). There were 50 fish in each tank, and at the end of the experiment, a sample of 10 fish was removed from each tank and their sperm cells examined for deformities. The data were the % of deformed sperm cells in each fish.
A t-test was then performed to test the H0 that the addition of Pb in the sea water did not affect the % of deformed sperm cells in fish. The result showed a t value of 11.48, with 9 df and a P value of < 0.001.
A biologist studying starlings wanted to know whether the mean mass of starlings differed according to different roosting situations. She was also interested in whether the mean mass of starlings altered over winter (Northern hemisphere) and whether the patterns amongst roosting situations were consistent throughout winter, therefore starlings were captured at the start (November) and end of winter (January). Ten starlings were captured from each roosting situation in each season, so in total, 80 birds were captured and weighed.
Open the starling data file.
Here is a modified example from Quinn and Keough (2002). Stehman and Meredith (1995) present data from an experiment that 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. With two factors, this experiment should be analyzed with a 2 factor (2 x 3) ANOVA.
Open the stehman data file.
The variable PH contains a list of pH values and is supposed to represent a factorial variable. However, because the contents of this variable are numbers, R initially treats them as numbers, and therefore considers the variable to be numeric rather than categorical. In order to force R to treat this variable as a factor (categorical) it is necessary to first convert this numeric variable into a factor.
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.
Open the quinn data file.
Confirm the need for a square root transformation, by examining boxplots and mean vs variance plots for both raw and transformed data. Note that square root transformation was selected because the data were counts (count data often includes values of zero - cannot compute log of zero).