Data Set 1A: Algal Photosynthesis vs. Salinity and Temperature

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1 Data Set A: Algal Photosynthesis vs. Salinity and Temperature Statistical setting These data are from a controlled experiment in which two quantitative variables were manipulated, to determine their effects on the quantitative response variable. Both explanatory variables therefore are of direct interest; there are no supplemental variables and there is no need for reduction of variables. The study used a balanced factorial randomized design. Three levels of one explanatory variable, and four levels of the other, were used in all possible combinations, with three replicates of each. In such a design the two manipulated variables are perfectly uncorrelated so that there is no confounding between them. The model used to analyze these data is fairly simple, including only linear terms for both explanatory variables and their interaction. (These data are a subset of the data from the actual study, chosen so that this simple model is suitable. Analysis of the full data set, presented in handout B, includes polynomial terms to model nonlinearity.) Background and Data Mastocarpus papillatus is a common intertidal red alga found along the west coast of North America from Baja California to Alaska. These data (supplied by Naomi Phillips, former Botany grad student) are for the photosynthesis / respiration ratio (P/R) of female gametophytes of M. papillatus, cultured at different salinities and s. The purpose was to explore possible explanations for the distributional limits of the alga in San Francisco Bay.

2 Data The data used in this handout were for three s (, and C) and four salinities (,,, and %). The table below contains the three observed values of the P/R ratio for each of the combinations of these levels of and. % % % % Data Exploration Graphical Since only a few discrete values of each explanatory variable were used, the three-way relationship among the variables can be graphed more easily than is often the case. In particular it is useful to plot the response variable vs. one explanatory variable, indicating the levels of the other explanatory variable by different symbols and/or line types. In the first graph to the right, with on the horizontal axis, there is no strong overall trend in P/R with changing. Considering the different s two differences can be seen. First, the P/R ratios tend to be highest at and lowest at. Second, the effect of is slightly different at different s: there is a weak negative trend with increasing at, essentially no trend at, and a weak positive trent at. These patterns have a different appearance in the second graph, with on the horizontal axis. The overall trend of decreasing P/R at higher is evident in the downward slope in the graph. The differences among the s in the direction of the relationship between P/R and, easily seen in the first graph, are seen in the second P/R P/R temp Data Set A: Algal Photosynthesis vs. Salinity and Temperature (rev. January, )

3 graph as a reversal in the ordering of the P/R values for the four salinities at vs. (% having the highest P/R, and % the lowest, at, and the reverse at ), and very little difference among the salinities at. Since there are only three variables, three-dimensional plots showing all variables quantitatively also are possible. Two such plots are shown here. The first shows all the. P/R... observations (with projection lines from the floor of the graph to help visualize the threedimensional locations of the points). The second (below) shows the means of the P/R for each of the combinations of and, connected as a surface.in both graphs, but perhaps. P/R... more easily in the surface plot of means, the patterns described above can again be seen. There is an overall downward trend with increasing (left to right), but much less so at higher (back of the graph) than at low (from of the graph), with the result that the trend in P/R relative to is positive (increasing from front to back) at the highest (right side) and negative (decreasing from front to back) at the lowest (left side). Data Set A: Algal Photosynthesis vs. Salinity and Temperature (rev. January, )

4 Summary Statistics With only observations per treatment only the simplest descriptive statistics make any sense; quartiles in particular are not useful. temp mean sd min median max range The means and medians show the trends described in the graphs: decreasing with increasing, more so at the lowest than the highest. With the exception of the %- treatment at which the three observations had identical values, the variability, measured by either the standard deviation or the range, also tends to decrease with increasing, but it does not have any clear relationship to. Inference Model The data exploration above indicates that there is an interaction between the two explanatory variables: the effect of either one of them on the P/R response variable varied depending on the value of the other explanatory variable. There was, though, no suggestion of nonlinearity. The model used in the analysis therefore included linear terms and the interaction: Y i = β + β S i + β T i + β S i T i + ε i with S =, T =, and Y = P/R ratio. Data Set A: Algal Photosynthesis vs. Salinity and Temperature (rev. January, )

5 Hypothesis Tests ANOVA table for model: Source DF SS MS F P Regression Residual Error..98 Total 9.9 R = 8.% The overall model thus is highly significant (P <.), and explains more than half the variance in P/R. There is strong evidence for an effect of the experimental factors on P/R. Tests for individual parameters: t tests: Predictor Coef SE T P Constant interaction.... F tests (ANOVA table): Source DF Seq. SS Adj. SS Adj. MS F P *temp All parameter estimates are significantly different from (P <.), and the coefficient for is highly significant (P <.). Note that the added-in-order SSes ( Seq. SS ) and added-last SSes ( Adj. SS ) are not identical: although and are perfectly uncorrelated with each other, they both are correlated with the interaction term. Estimation The model: The estimated (fitted) model is: Ŷ i =.8.8S i.t i +.S i T i Because of the interaction term it is not possible to quantify the effect of either experimental factor by itself: the effect of one factor depends on the level of the other. We can instead see how each relationship varies over the range of values of the other factor. For instance, at = %, the estimated mean P/R is a linear function of, with coefficients gotten by plugging the value of into the model above: Data Set A: Algal Photosynthesis vs. Salinity and Temperature (rev. January, )

6 Ŷ i =.8 (.8 ).T i + (. )T i = (.8.9) + (. +.9)T i =..98T i Similar calculations for the other levels of yield the following equations: = : Ŷ i =. -.T i : Ŷ i = T i : Ŷ i =. -.8T i : Ŷ i =. -.89T i The estimated effect of is negative but becomes much less steep as increases, with the intercept similarly decreasing. The same analysis in terms of how the P/R to relationship varies with yields the following set of equations: = : Ŷ i =.8 -.S i : Ŷ i =. -.8S i : Ŷ i =.8 +.S i These equations show that the effect of on P/R varied from strongly negative at the lowest to slightly positive at the highest. Plotting these two sets of equations produces graphs of the fitted model corresponding to the first two scatterplots of the data shown earlier, as below: fits 9 8 temp fits Data Set A: Algal Photosynthesis vs. Salinity and Temperature (rev. January, )

7 Alternatively, the full fitted model can be plotted as a surface a twisted plane in a threedimensional plot: fits Comparison of this graph with the surface plot of the means shown earlier suggests that this model adequately describes the main pattern of the data, though obviously it doesn t reproduce the sawtooth variability around the overall trends with increasing. Confidence Intervals for the treatment combinations: For the combinations of values of and used in the study, the estimated P/R ratios with individual 9% CIs (for the mean Y) are given in the table on the next page. % Ŷ :.88 CI:., 8..9.,...,.88 %.8.98,. %.9.8,..8.,...98,..8.,..9.8,.9 %.89.99,.98..8,...9,. Simultaneous inference: The t critical value used for these individual 9% intervals was t*.9, =.9. To get CIs with simultaneous confidence level of 9%, we would replace this value with the Bonferroni-adjusted t critical value, t* -./(x), = t*.999, =.888. The intervals thus would be. times as wide as the individual intervals above. Data Set A: Algal Photosynthesis vs. Salinity and Temperature (rev. January, )

8 Diagnostics Since this was a designed, controlled experiment, diagnostic assessment of the adequacy of the regression model can be fairly simple. In particular no observations will have particularly high leverage, so sophisticated investigation of possibly influential observations is not required. Residual Plots Residuals vs. variables: Scatterplot of RESI vs FITS RESI FITS Scatterplot of RESI vs Scatterplot of RESI vs temp RESI RESI temp 8 There is no suggestion of nonlinearity in these residual plots. The variability is not as even as might be desired. In particular, there is less variability in the residuals for the lowest values of the fits (corresponding to observations at ) than for higher values of fits (lower s): the standard deviation of the residuals is.8 for observations at compared with. and. at and. A Levene s test comparing the variances of these three sets of residuals, however, is not significant (P =.). Data Set A: Algal Photosynthesis vs. Salinity and Temperature (rev. January, ) 8

9 Distribution of residuals: Histogram of RESI 99 Probability Plot of RESI Normal Frequency Percent RESI RESI This distribution is mildly skewed, with two particularly large positive values (from the observations with = %, =, P/R = 8., and with = %, =, P/R = 8.). Given the sample size of, though, this mild non-normality does not invalidate the analysis. Lack-of-Fit Test Since there are true replicates multiple observations with the same values of the explanatory variables a lack-of-fit test can be conducted, partitioning the error SS into the variability among the replicates ( pure error ) and the variability of the treatment means from the regression fits. Regression Residual Error..98 Lack of Fit Pure Error.99. Total 9.9 This test is clearly non-significant, suggesting the linear + interaction model is appropriate for these data. Conclusion from Diagnostics I see no substantial problems, and conclude that the fitted model is an adequate description of the data. Data Set A: Algal Photosynthesis vs. Salinity and Temperature (rev. January, ) 9

10 Conclusions Temperature and clearly do affect the P/R ratio of this alga, at least under these experimental conditions and for the population of algae from which the experimental units were collected. For the most part both higher and higher are detrimental (produce less photosynthesis relative to respiration). These effects, however, are less than additive: the decrease in P/R caused by an increase in one of the variables is less when the other variables is at a high level than when it is lower. At the highest, indeed, the increasing appears to be beneficial: estimated P/R increases with increasing. Assuming that a higher ratio of photosynthesis to respiration corresponds to higher net primary productivity by the alga, it presumably also corresponds to a higher potential rate of increase of the algal population. Thus the results of this experiment suggest that the algal population should thrive in cool, low conditions, and could be physiologically restricted from warmer and/or more saline conditions, with probably being the more important factor. The interaction term, though, suggests that the P/R ratio might also reach high levels in water both warmer and more saline than any used in this study. Unfortunately I don t know anything about the distribution of water and in San Francisco Bay and therefore have no idea whether these conclusions match the actual distribution of the alga. They do appear to conflict from a published study of photosynthesis in M. papillatus which reports maximal photosynthesis at C (with more southern populations having higher optimal s), but obviously the ratio of photosynthesis to respiration could have a different relationship to than does photosynthesis alone. There also are reasons to question whether the response variable in the data as given me by Naomi is entirely appropriate as an indicator of either physiological or population vigor of this alga. The difference between photosynthesis and respiration, rather than their ratio, more properly measures net productivity. It is possible that if, for instance, both rates increase with increasing, the difference might also increase with while the ratio decreases. Furthermore, M. papillatus lives in the intertidal zone and thus spends substantial amounts of time out of water, subjected both to dessication and to s higher and more variable than water s. The experimental conditions in the present study therefore probably do not include the most physiological stressful conditions the alga encounters in nature. Finally, of course, the distribution of the alga may be determined by factors other than the effects of and on photosynthesis and respiration, whether in the water or not. In short, the experiment gives a clear, presumably reliable description of the effects directly studied, while the relevance of these effects to the ecology of the alga is uncertain. Data Set A: Algal Photosynthesis vs. Salinity and Temperature (rev. January, )

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