Chapter 13: Analysis of variance for two-way classifications

Transcription

1 Chapter 1: Analysis of variance for two-way classifications Pygmalion was a king of Cyprus who sculpted a figure of the ideal woman and then fell in love with the sculpture. It also refers for the situation in which high expectations placed on individuals by teachers or supervisors often results in improved performance by students or subordinates. Eden 1 speculated that in most quantitative examples of the Pygmalion effect which compared two groups of subjects (one with high expectations, and the other without), there were also reduced expectations placed on the control group. This contrast between high and low expectations may be exaggerate the Pygmalion effect. Eden conducted an experiment that attempted to more fairly isolate the Pygmalion effect by using ten companies of soldiers that were to undergo basic training. Each company consisted of two or three platoons; one platoon in each company was randomly selected to receive the Pygmalion treatment and the remaining platoons were to receive a control treatment. Prior to assuming command, each platoon leader met with an army psychologist that described a nonexistent battery of tests taken by the platoon members. If the platoon was a Pygmalion treatment platoon, the psychologist reported that the tests predicted superior performance for the platoon. At the end of training, members of each platoon took a test that measured their ability to operate weapons and answer questions about their use. The platoon mean scores are the response, and if the Pygmalion effect is real, then it is expected that the Pygmalion treatment platoons will tend to score higher than the control platoons. There are two sources of potential differences in platoon means: treatment (Pygmalion versus control) and company (a company is a cohesive unit and the platoons within the company tend to be treated alike with respect to housing, meals and so on). Company is best thought of as a blocking factor as the assignment of platoon to company was not under the control of Eden and the possible effect of a platoon being assigned to specific company (e.g. A company) is of no lasting interest. The only purpose of modeling the effect of company is to control for differences among companies so that the Pygmalion effect may be more precisely estimated. The experimental design is a randomized block design. The factor of principal interest is the Pygmalion factor, and it is a fixed factor. Companies, the second factor may be treated as either a fixed factor (for simplicity) or a random factor, though technically, the treatment as a random factor is arguably more realistic. 1 Eden, D. 1990, Pygmalion effects without interpersonal contrast effects, J. Appl. Psychol. 75(4), Company does not perfectly fit the standard definition of a block as a factor created to remove the effects of nuisance variables on the response. Each block ought to be as homogeneous as possible with respect to the nuisance variables so that the comparison of treatments (within block) is not clouded by the nuisance variables. A random factor is one for which the realized levels have been randomly drawn from a population of levels. Company is logically a random factor if it is reasonable to envision a population of infinitely many 106

2 The experimental units are platoons, not individual Treatments soldiers since it was platoons that were assigned to Company Pygmalion Control treatment group and block. Scores on individual soldiers are averaged, then discarded. The data are shown in the table to the right. There is one observation for each combination of company and the Pygmalion treatment, and usually two observations for each combination of company and the control treatment. The experiment is not balanced The data is presented in a row by column (two-way) table as this format is logical: each row may be systematically different from all other rows because the data originates from a single level of the row factor; likewise, each column may be systematically different from the other (if the Pygmalion effect is real). The number of levels of the row factor is denoted by r (hence r = 10) and the number of levels of the column variable is denoted by c denote (hence c = ). There are rc treatment combinations; with reference to the tabular lay-out of the data it is said that there are rc cells. Additive and non-additive models for two-way tables When there are two explanatory variables that are both factors, the data may be viewed as a two-way (rows and columns) table where rows and columns each correspond to a factor. In the Pygmalion study, each row corresponds to a level of company (with 10 levels), and each column corresponds to a treatment ( levels). There are two broad models that are predominantly used with two factors: additive and non-additive models. An additive model assumes that there is no interaction between the row and column factors. Consequently, the effect of one level of one factor (e.g., the row factor) is the same at all levels of the second factor. The effect of factor A is completely unrelated to the levels of B; likewise, the effect of factor B is completely unrelated to factor A. The term additive is used because the model estimate of the expected response at level a of factor A and level b of B is the sum of the effect of A at level a and the effect of B company effects and that the distribution of the effects in normal or nearly so. 107

3 at level b. If the model were non-additive, the the sum of the effects of A at a and B at b generally is not the sum of the two effects. The non-additive model contains terms that allow for different effects of level of A depending on the level of B; thus, the non-additive model is what has previously been described as an interaction model. The additive model for the Pygmalion experiment is µ(y x) = β 0 + β 1 x Pygmalion + β x Co. + + β 10 x Co.10 where 1, if treatment is Pygmalion x Pygmalion = 0, if treatment is control, x Co. = x Co.10 = 1, if company is 0, if company is not,. 1, if company is 10 0, if company is not 10. The reference level for the Pygamlion treatment is the control level and the reference level for company is company 1. Each model coefficient (besides β 0 ) is then the difference between the reference level and some other level. The model for each specific combination of levels is shown in Table 1. Table 1: Additive model for the Pygmalion experiment. Treatments Treatment effects Company Pygmalion Control Pygmalion Control 1 β 0 + β 1 β 0 β 1 β 0 + β 1 + β β 0 + β β 1 β 0 + β 1 + β β 0 + β β 1 4 β 0 + β 1 + β 4 β 0 + β 4 β 1 5 β 0 + β 1 + β 5 β 0 + β 5 β 1 6 β 0 + β 1 + β 6 β 0 + β 6 β 1 7 β 0 + β 1 + β 7 β 0 + β 7 β 1 8 β 0 + β 1 + β 8 β 0 + β 8 β 1 9 β 0 + β 1 + β 9 β 0 + β 9 β 1 10 β 0 + β 1 + β 10 β 0 + β 10 β 1 Thus, β 1 is the Pygmalion effect (the difference in expected response between the control and Pygmalion treatments (µ Pygmalion µ control ), β is the difference in mean platoon score 108

4 between company and company 1 (µ µ 1 ), and so on. The effect of the factors µ(y x) are independent of each other. Consequently, no matter which company is scrutinized, the Pygmalion effect is the same. If the additive model fails to fit well compared to the non-additive (interaction model), then all references to the Pygmalion effect must be stated with respect to a particular company. Generically, let r denote the number of levels of the row factor (hence r = 10) and c denote the number of levels of the column variable (hence c = ). Then, the number of parameters in the additive model is p = 1 + (r 1) + (c 1) = = 11. The saturated or nonadditive model The alternative model to the additive model is the saturated, nonadditive model. 4 The saturated model specifies that the row and column factors interact and in doing so, implies that the effect of some of factor levels are not the same at each level of the other factor. The term called saturated is used because no additional parameters can be introduced into the model. Specifically, the interaction model contains as many parameters as there are cells or treatment combinations (0 = 10 cells). The interaction variables are set up by forming the product or each row factor indicator variable (there are r 1) with each column factor indicator variable (there are c 1). Using the Pygmalion experiment as an example, the interaction variables are 1, if treatment is Pygmalion and company is x 11 =x 1 x = 0, otherwise. 1, if treatment is Pygmalion and company is 10 x 19 =x 1 x 10 = 0, otherwise The saturated model contains p = 1 + r 1 + c 1 + (r 1)(c 1) = r + (c 1)(1 + r 1) = r + (c 1)r = rc 4 The model µ(y x) = β 0 is arguably an alternative as well. 109

5 parameters. Using the indicator variables set up above, the saturated model is shown for each cell below. Table : The saturated model or nonadditive model specified in terms of regression parameters for the Pygmalion experiment. Treatments Company Pygmalion Control Pygmalion control 1 β 0 + β 1 β 0 β 1 β 0 + β 1 + β + β 11 β 0 + β β 1 + β 11 β 0 + β 1 + β + β 1 β 0 + β β 1 + β 1 4 β 0 + β 1 + β 4 + β 1 β 0 + β 4 β 1 + β 1 5 β 0 + β 1 + β 5 + β 14 β 0 + β 5 β 1 + β 14 6 β 0 + β 1 + β 6 + β 15 β 0 + β 6 β 1 + β 15 7 β 0 + β 1 + β 7 + β 16 β 0 + β 7 β 1 + β 15 8 β 0 + β 1 + β 8 + β 17 β 0 + β 8 β 1 + β 17 9 β 0 + β 1 + β 9 + β 18 β 0 + β 9 β 1 + β β 0 + β 1 + β 10 + β 19 β 0 + β 10 β 1 + β 19 Every cell contains a unique sum of parameters, and so the table could be revised by writing µ 1 = β 0 + β 1, µ = β 0, µ = β 0 + β 1 + β + β 1,..., µ 0 = β 0 + β 10. Thus, there is one unique parameter for every cell. A cell mean is the mean of all observations obtained at a particular treatment combination (or cell). For example, for Company 1, control platoon, the cell mean is µ = = 66.. The mathematical proof is not simple, but with two-way tables, the saturated model estimate of µ ij (fit by multiple linear regression) is equal to the sample mean of the n ij observations belonging to the row i column j cell. For brevity, µ ij = y ij. The term cell means model is sometimes used for the saturated (or non-additive) model because the estimate for each cell is unconstrained (doesn t depend on any other observations besides those belonging to the cell). Let y ijk denote the platoon mean for level i of company, i = 1,..., 10, and level j of treatment, j = 1,, and replicate k (= 1 or ). n ij identifies the number of replicates, and so n ij is 1 whenever j = 1 (Pygmalion treatment), and n ij is for the controls (except 110

6 company ). If n ij = 1, then ŷ ij = µ ij = y ij = y ij ; in other words, the model fits the data for cell ij with zero error. The estimate of σ the residual variance is σ = 1 n p r i=1 n c ij (y ijk ŷ ij ). j=1 k=1 where ŷ ij = µ ij is both the fitted value obtained from the fitted regression model and the cell mean for row i and column j. Since there is a single observation. For this example, there are n p = 9 0 = 9 are the degrees of freedom for error. Another way of looking at the degrees of freedom is that there are rc = 0 cells; of these 11 have a single observation. If n ij = 1, then the estimate will be exactly equal to the observation and the residual error will be zero obviously incorrect (it s not reasonable to expect the model to fit another data set with n ij > 1 without error). The only cells that can be used to estimate error are those with more than one observation, and there are n rc = 9 0 = 9 of these cells that are useful for estimating error. Hence, df = n rc = 0. A strategy for analyzing two-way tables with several observations per cell The fixed effects analysis of variance is approached as a multiple regression analysis in which backwards elimination determines the importance of the factors. 1. Begin with graphically-based initial exploration, and determine if there are outliers, and if transformations are needed.. Fit a rich model with interactions (the saturated model), and examine model assumptions (concentrating on the constant variance assumption, and whether there are outliers).. Use the extra-sums-of-squares F -test to determine if interaction can be eliminated from the model. If interaction is significant, then estimate µ ij and σ(µ ij ) using µ ij = y ij and σ(y ij ) = σ/n ij for each i and j where σ is the residual standard error obtained from the fitted regression model If interaction terms are not needed, then test whether the additive effects of the row factor are zero, and whether the additive effects of the column factor are zero. In other words, test the significance of the row and column factors unless one factor is a 5 There are other ways of computing the standard errors, but finding a simpler method is not easy. 111

7 blocking factor. If a factor is significant, then particular comparisons can be carried out. For example, estimate the differences in expected response for different treatments (when interaction is found to be present) or different levels of factors (when interaction is not present). Blocking There is little point in testing the significance of a blocking factor because the levels of the blocking factor are (in principal) chosen so that the response variable is as similar as possible within block (to maximize the sensitivity of significance test for the other factor). The question of whether the response variable differs among blocks is not a important question. Moreover, if the blocking factor is omitted from the model, little is gained. Similarly, there is little scientific motivation in most instances to consider a model involving interaction between the blocking and other factor. The reason is again because the levels of the blocking factor are (in principal) chosen so that the response variable is a similar as possible within block, a rationale that does not suggest that interaction should be present. The analysis of variance F -test for additivity This test is nothing more than an extra-sums-of-squares F -test that compares the unconstrained model (both factors and their interaction) against the constrained alternative (both factors, no interaction.) The constrained model is nested within the unconstrained model because if the all the interaction parameters are equal to 0, then the nonadditive (unconstrained) model reduces to the additive model. The extra-sums-of-squares F -test The extra-sums-of-squares F -test is used to formally compare the fit of two competing models when one model is a constrained version of the another. It was presented in Chapter 6 within the context of the one-way analysis of variance and in Chapter 10 for multiple regression problems. The objective is to compare the lack-of-fit of the additive model containing p additive = 1 + r 1 + c 1 = r + c 1 model parameters to the lack-of-fit of the non-additive model containing p nonadditive = 1 + (r 1) + (c 1) + (r 1)(c 1) = rc model parameters. 11

8 Let σ nonadditive = SSR nonadditive n rc denote the estimated residual variance obtained from the nonadditive model (the unconstrained or saturated model). The hypotheses are H 0 : β i = β i+1 = = β i+k = 0 versus H a : at least one of β i, β i+1,..., β i+k is not 0. where β i = β i+1,..., β i+k are the k = (r 1)(c 1) interaction parameters. The test statistic is F = SSR additive SSR nonadditive (r 1)(c 1) σ nonadditive = MS lack-of-fit. σ nonadditive where (r 1)(c 1) is the difference in number of model parameter between the constrained and unconstrained models, and MS lack-of-fit = (SSR additive SSR nonadditive )/(r 1)(c 1). If H 0 is correct, then F F (r 1)(c 1),n rc As always with F -statistics, p-value = P (F f H 0 ), where f is the observed value of the test statistic. A p-value obtained from this test will be accurate if the random error terms ε ij are at least approximately independent and identically distributed N(0, σ). This assumption must be investigated by residual plots which check for non-constant variance. A quantile-quantile plot is used to check for approximate normality. The residuals used in this analysis are the residuals from the nonadditive model, since σ is estimated using the residual mean square error from the nonadditive model. Studentized residuals Fitted values

9 The figure above and right is a residual plot using residuals from the nonadditive model. There is no concern regarding the assumption of constant variance. The figure to the right plots the platoon means and the fitted values obtained from the nonadditive model. The figure reveals that there is some consistency among the differences between the Pygmalion and control means and, at the same time, there are two companies ( and 9) that differ from the general trend. A formal test of significance is necessary. An extra-sums-of-squares test testing whether the interaction terms are non-zero is shown in Table. There s no evidence supporting interaction between company and treatment. Platoon mean Company Table : The extra-sums-of-squares F -test for interaction between company and treatment. Source of Residual Sum variation of squares d.f. Mean square F -statistic p-value Additive model Lack-of-fit Nonadditive model Table 4 below shows that, after accounting for differences between companies, there is convincing evidence that the pygmalion effect is real. The estimated effect is β 1 = 7. and a 95% confidence interval for β 1 is [1.801, 1.69]. Table 4: The extra-sums-of-squares F -test for the significance of treatment after accounting for the effects of company. Source of Residual Sum variation of squares d.f. Mean square F -statistic p-value Company-only Lack-of-fit Additive model The analysis is summarized in Table 5. The sums-of-squares are computed by first removing the treatment factor from the additive model to compute the sums-of-squares shown on the 114

10 treatment line; the sums-of-squares for company was computed from the difference of residual sums-of-squares from the model with only company as a factor and the model with only an intercept. As discussed above, the test for company effects is not particularly interesting since company is a blocking factor. However, the test is conducted automatically using the R function call summary(aov(score~company+treat)). Some care is needed since the function call summary(aov(score~treat+company)) will compute the sums-of-squares by removing first company from the additive model and then treatment. This ordering is incorrect for a two-way table when one factor is a blocking variable since the treatment factor must be tested while accounting for the blocking factor. 6 For comparative purposes, the analysis of variance table obtained from the incorrect function call is shown in Table 6. The tables are similar but not equivalent because the data are not balanced. If the data were balanced (i.e., n ij were the same for each cell), then the order in which the terms are dropped does not substantially affect the test. Table 5: Analysis of variance table for the pygmalion experiment. Source of Residual Sum variation of squares d.f. Mean square F -statistic p-value Company Treatment Residual error Table 6: Incorrect analysis of variance table for the pygmalion experiment. The sums-ofsquares were obtained by dropping company from the additive model followed by treatment. Source of Residual Sum variation of squares d.f. Mean square F -statistic p-value Treatment Company Residual error Seaweed in the intertidal zone To study the influence of grazers on regeneration rates of seaweed in the intertidal zone, Olsen 7 scraped rocks free of seaweed and observed the amount of regeneration over time when certain grazers were excluded. One hundred 1 cm plots (exclosures) were constructed 6 There is no real value to testing for the significance of the blocking variable. 7 (A. Olsen, Evolutionary and Ecological Interactions Affecting Seaweeds, Ph.D Thesis. Oregon State U. 199.) 115

11 by mounting nets on a frame bolted to the rock substrate. All plots had frames to eliminate confounding with the possible effect of the frames on feeding preference. The grazers were 1. L - limpets (an invertebrate). f - small fish. F - big fish Each plot received one of 6 treatments: 1. LfF: all three grazers were allowed access. ff: fish allowed access (limpets excluded by surrounding the plot with a caustic paint). Lf: Limpets and small fish allowed access (a coarse net excluded large fish) 4. f: small fish allowed access (paint and coarse net) 5. L: limpets allowed access (fine net) 6. C: (control) limpets, small and large fish excluded The table shows the treatments: Limpets absent Limpets present Small fish Small fish Large fish absent present absent present absent C f L Lf present ff LfF In principle, three factors might be identified: limpets (present and absent), small fish (present and absent), and large fish (present and absent). The design is similar to a factorial design that combines each level of each factor with every other level. However, if this were a factorial design, then there would be = 8 treatments. It was not possible to form all 8 combinations; for instance, it was not feasible to exclude small fish and allow large fish in the enclosures. Instead, the experiment is treated as a two-way analysis of variance using a single treatment factor with 6 levels and a blocking factor corresponding to inter-tidal environment. Because the intertidal zone is a highly variable environment, the treatments were replicated in eight blocks, each containing 1 plots. Within block, the six levels were randomly allocated to the 1 plots; each treatment level is replicated twice within block. The blocks are 1. Block 1: below high tide, exposed to heavy surf 116

12 . Block : below high tide, protected from heavy surf. Block : Mid-tide, exposed 4. Block 4: Mid-tide, protected 5. Block 5: Low tide, exposed 6. Block 6: Low tide, protected 7. Block 7: On a near-vertical rock wall, mid-tide level and exposed 8. Block 8: On a near-vertical rock wall, low tide level and protected The experiment is a randomized block experiment since treatment levels were randomly allocated to experimental units (the plots) within each block. Because there where two replications of each treatment in each block, the design is balanced. After four weeks, Olsen estimated regenerating seaweed cover by positioning a metal sheet with 100 holes over each plot. The percentage of holes that were positioned over regenerating seaweed was determined. Objectives 1. Determine the impacts of the three different grazers on seaweed regeneration rates.. Determine which grazer consumes the most seaweed.. Determine if different grazers affect each other. 4. Determine if grazing effects are the same in all microhabitats. The data are shown in a coplot below. The R function call used to construct the coplot is coplot(y~treat Blk,ylab="Percent regeneration",xlab="treatment level",pch=16). Differences among treatments and blocks appear to be substantial. The residuals from the nonadditive model are shown to the right and clearly reveal a substantial problem with nonconstant variance (the variances of the residuals about the fitted values is largest when the fitted value is near 50%). The logit transformation was used to reduce non-constant variance and to eliminate the upper and lower bounds (100% and 0%) on the responses. The logit transformation is ( ) y logit(y) = log. 100 y 117

13 Given : Blk Percent regeneration B B B1 C f ff L Lf LfF B4 B5 B8 B7 B6 C f ff L Lf LfF Residuals C f ff L Lf LfF Treatment level Fitted values The figures below show the result of the transformation on the response variable and the residuals about the nonadditive model. A normal probability plot of the residuals reveals no evidence that the distribution of residuals departs from a normal distribution. Given : Blk B1 B B B4 B5 B6 B7 B8 Percent regeneration 4 0 C f ff L Lf LfF C f ff L Lf LfF 4 0 Residuals C f ff L Lf LfF Treatment level Fitted values There are obvious large differences (judging from the Figure above) attributable to the treatment and block. The analysis of variance table (Table 7) shows that there s sufficiently little 118

14 evidence of interaction to justify adopting the nonadditive model. The test for interaction provides information toward answering the question of whether grazing effects are the same in all microhabitats; specifically, it is concluded that there is insufficient evidence to conclude that grazing effects differ among the eight microhabitats. Table 7: Analysis of variance table for the nonadditive model, seaweed grazers experiment. R =.98, n = 96, σ =.550. Source of Residual Sum variation of squares d.f. Mean square F -statistic p-value Blocks <.0001 Treatment <.0001 Interaction Residual error Total The additive model is summarized in Table 9. Because the experiment is balanced (n ij = for every i and j), the sums-of-squares attributable to each main effect is the same as in Table 7. If the experiment (or data) are not balanced, then the sums-of-squares depends on what terms are in the model. The F -statistics are slightly different because the denominator of the statistic ( σ ) differs between the nonadditive and additive model. Table 8: Analysis of variance table for the additive model, seaweed grazers experiment. R =.85, n = 96, σ = Source of Residual Sum variation of squares d.f. Mean square F -statistic p-value Blocks <.0001 Treatment <.0001 Residual error Total Table 9 shows the fitted values from the additive model. The row means (i.e., the mean prediction for all observations in a particular block) are shown in the rightmost column. 8 The column means (bottom row) are the mean fitted values for all observations in a particular treatment group. The row means (rightmost column) are the mean fitted values for all observations in a particular block. The column means will be used to compare the effects of each type of grazer and determine if the grazers affect each other. 8 The ith row mean is µ i = 1 c c µ ij, j=1 119

15 Table 9: Fitted values (on the logit scale) derived from the additive model. Treatment Blocks C f ff L Lf LfF Mean Mean Recall 9 that a linear combination of means µ 1, µ,..., µ I is a sum γ = C 1 µ 1 + C µ + + C I µ I where C 1, C,, C I is a set of known constants. If 0 = C 1 + C + + C I, then the linear combination is called a contrast. Since an additive model was adopted, the effect of limpets 10 can be estimated by comparing the mean response for the three treatments that allowed limpets to the mean response for the three treatments that excluded limpets using a contrast of treatment means given by γ = µ LfF + µ Lf + µ L µ ff + µ f + µ C. Generally, a test of the hypothesis H 0 : γ = 0 versus H a : γ 0, uses the t-statistic where the standard error of γ is t = γ σ( γ) σ( γ) = σ C 1 n C I n I, 9 see Chapter 8 10 One objective was to determine which grazer consumes the most seaweed; to answer the question, it s necessary to estimate the effect of each grazer. 10

16 and σ is the residual standard error associated with the final (adopted) model. Specifically, If H 0 : γ = 0 is true, then σ = SS Add n I I ni i=1 j=1 = (y ij ŷ ij ). n p t = γ σ( γ) t n p. Unusually large or small values of t are consistent with H a : γ 0 and evidence against H 0, and so p-value = P (T n p t ). From Table 7, SS Add = is the residual sums-of-squares from the adopted model, in this case, the additive model. Then, σ =.599 =.59. The denominator sample sizes are number of observations that are used to estimate the treatment means. In this case, n i = 16 = 8 for each treatment i = 1,..., 6. The estimated effect of limpets is computed using the treatment (or column) means µ LfF =.7,..., µ C =.18 from Table 9. Using the column means is consistent with the additive model as the additive model specifies that the treatment effect is the same in all blocks. Maximally precise treatment mean estimates are obtained by averaging all the cell means corresponding to a particular treatment. Consequently, µ LfF + µ Lf + µ L µ ff + µ f + µ C = = The estimated variance of the treatment mean contrast is t σ ( γ) = σ Ci n i=1 i t = σ Ci 16 and = i=1 ( [ 1 ] + [ ] 1 + = =.01496, [ ] [ [ + ] 1 [ + ] 1 ] ) σ( γ) = σ ( γ) = =.1. The t-statistic is t = γ σ( γ) = =

17 and p-value = P (T ) < There is convincing evidence that limpets affect the regeneration of seaweed. A 95% confidence interval for γ is γ ± t σ( γ) = 1.88 ± = [.07, 1.58], where.011 is the.05 quantile of the t-distribution with n p = 48 degrees of freedom. The corresponding contrast to test for an effect due to small fish uses the contrast µ LfF + µ ff + µ Lf + µ f 4 µ L + µ C = 4 = The estimated variance of the treatment mean contrast is and The t-statistic is σ ( γ) = σ = t i=1 Ci n i ( [1 ] + 4 [ ] = =.0168, [ ] σ( γ) = σ ( γ) =.0168 =.197. t = γ σ( γ) = = [ ] [ [ + 4 ] 1 ] ) and p-value = P (t ) < There is convincing evidence that small fish affect the regeneration of seaweed. The question of whether different grazers affect each other is best addressed by contrasts. For example, to investigate whether limpets are affected by small fish, I will compare the differences between limpets present and absent when small fish are present, versus when small fish are absent. A contrast of these means is The contrast estimate is µ LFf µ ff + + µ Lf µ f µ L µ C and the estimated standard error of the contrast is σ( γ) =.59. =.095, The test statistic and p-value are t =.095/.59 =.610 and p-value = P (T 48 >.610) =.4 which shows that there is no evidence that small fish affect limpets, and likewise no evidence that limpets affect small fish. 1