# Agenda

## To Dos

• Check with Jon about Fonts for figures.
• Are t.b.s really as variable as all that? From newprovisioning_florida.xls: RWL 55 has mass of 2300 (female), whereas another with RWL 55 has a mass of 3100 (male)?

## Announcements

Jon Hastings is on a NASCAR vacation, but wants to meet on Wednesday, 5/27, at 1:00. Hope that works for you....

Jon Hastings's rough drafts for this manuscript, destined for the Florida Entomologist

Todo items for this meeting:

• Katie agreed to look at the Florida Entomologist requirements for authors?
• Here are the directions from the Florida Entomologist requirements for tables and figures:
• Submit all figures and photographs as .jpg or .tiff files. All figures and tables must be referenced in the text with Arabic numerals in the order in which they appear in the text. Table footnotes are written below the table and indicated as superscript Arabic numerals. Table legend in uppercase. All captions for figures are listed together on a separate page. All illustrations must be complete and final. Make a composite figure if numerous line drawings or photographs are needed; do not combine photographs and drawings in the same figure.
• Figure 3: Katie will produce them (please do one with symbol by location, and one with symbol by size -- small, medium, large -- per Jon's request); Grayson will comment on it (rough version here)
• Wasp RWL to Mass: Katie will finalize, and comment
• Cicada RWL to Mass: Grayson and I will continue to work on the step-wise regression, then Katie will produce the final figure. Grayson will comment.
• Stair-step figure: I will finalize the model, and get the approximate SEs. I will comment.
• Crude opportunistic statistic: I will calculate and comment.

Last time we were looking at some of the results of some propositions in materials Jon sent from the literature. Katie was able to contest the notion that there was a change in the regression of RWL versus Mass for wasps: we don't see the hypothesized decrease in RWL growth with increasing Mass here's an animation of the results... In fact, the relationship seemed to be in the opposite direction.

Grayson was able to determine that the Dow data provided evidence consistent with about a 25% mass conversion relationship.

I presented some information related to the non-linear regression standard errors, which we will need to present along with our models.

## Contributions to the Manuscript So Far

### Data sets

We've verified that we're using the proper data sets (conforming to those that Jon wants to use) in the following:

### Analyses/Figures

• Use the breakpoints of our stair-step model and the census provided by the wasps in St. John's (use data set FLORIDA.xls) to calculate the probability that the largest class of wasps could be choosing their cicadas opportunistically. We compute the probability that the large wasps, capable of handling any cicada, would be hunting opportunistically. As background frequencies of the cicada classes we take the empirical frequencies obtained by censusing all female wasps and their prey during the period [Jon?].
 Cicadas size category small medium large N small wasps 19-28.37 47 0 0 47 medium wasps 28.37-33.66 14 67 14 95 large wasps 33.66- 0 1 29 30 totals 61 68 43 172

The probability that we would find a distribution as extreme or more so than that actually obtained is equal to the independent probabilities ${\displaystyle p=binomial(0,30,61/172)*\sum _{i=0}^{1}binomial(i,30,68/111)\approx 4.2\times 10^{-17}}$

• Table 3 of the Grant paper: We'll create two versions of this, one with symbol style based on location, the other with symbol style based on cicada species. For the moment, here's a rough version of the former:

(Click to enlarge the figure.) The lines have slopes 2.5, 2, 1.5, 1, .5, and .25. Katie is making the final plots, and Grayson will provide commentary.

• The graphic of the model of the wasp RWL/Mass conversion is available from Katie. She's also going to be providing a second figure, showing the linear regression of the log-transformed variables and the linear model, the non-linear model formula, standard errors, etc. She'll do the write up. Here is the preliminary non-linear model plotted against the data:

• The graphic of the model of the cicada RWL/Mass conversion is available from Katie. Grayson and I are working on the model, using only small, medium, and large classes. We will provide a table of the linear regression results of the log-transformed variables and the linear model, the non-linear model formula, standard errors, etc. Grayson will do the write up. Here is the preliminary non-linear model plotted against the data:

#### Analysis:

Data files:

Here's my take:

• xlispstat came up with two models: either lnRWL alone, or lnRWL with the middle*lnRWL terms. There is one rule of thumb which states that you don't throw an interaction term in unless you also throw in the intercept term: in other words, you wouldn't put in middle*lnRWL without middle also. Since we are choosing between these two, we should choose middle first.
• Using R, we see, through both add1 and drop1, that the conclusion is that it's certain that we need lnRWL, and then it's a toss-up between medium and medium*lnRWL interaction term. Since it's virtually a toss-up, I'd say that we go with the three term model

Fit <- lm(lnMass ~ lnRWL + medium)

Another advantage of this model is that the slope is almost exactly 3, as we would expect intuitively, and the introduction of "medium" means only an intercept change, rather than a slope change.

Results:

Call:
lm(formula = lnMass ~ lnRWL + medium)

Residuals:
Min        1Q    Median        3Q       Max
-0.309325 -0.081741  0.001887  0.090330  0.273934

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.08275    0.14327  -28.50  < 2e-16 ***
lnRWL        3.02359    0.04220   71.66  < 2e-16 ***
medium       0.22861    0.03008    7.60 3.87e-12 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1137 on 140 degrees of freedom
Multiple R-squared: 0.9829,     Adjusted R-squared: 0.9827
F-statistic:  4033 on 2 and 140 DF,  p-value: < 2.2e-16

R-code:

species<-ctable[,1]
mass<-ctable[,2]
RWL<-ctable[,3]
lnMass <- log(mass)
lnRWL <- log(RWL)
levels(species) <- c("s", "s", "l", "m") # Grayson: T.b. comes before T.g.
Fit <- lm(lnMass ~ lnRWL + factor(species) + factor(species)*lnRWL)
lm1 <- lm(lnMass~1, data=ctable)
Fit <- lm(lnMass ~ lnRWL + factor(species)*lnRWL) # automatically does constant plus cross terms
summary(Fit)

# Let's try some new data, in which I add the indicator variables:
lnMass<-ctable[,1]
lnRWL<-ctable[,2]
small<-ctable[,3]
medium<-ctable[,4]
large<-ctable[,5]

mediumCross <-medium*lnRWL
largeCross <-large*lnRWL

lm1 <- lm(lnMass~1,data=ctable)
add1(lm1, ~ lnRWL + medium + large  + mediumCross + largeCross)

lm2 <- lm(lnMass~ lnRWL)
summary(lm2)
add1(lm2, ~ lnRWL + medium + large  + mediumCross + largeCross)

lm3 <- lm(lnMass~ lnRWL + medium)
summary(lm3)
add1(lm3, ~ lnRWL + medium + large + mediumCross + largeCross)

Fit <- lm(lnMass ~ lnRWL + medium + large  + mediumCross + largeCross)
summary(Fit)
drop1(Fit, test="F")

Fit <- lm(lnMass ~ lnRWL + medium + large  + mediumCross)
summary(Fit)
drop1(Fit, test="F")

Fit <- lm(lnMass ~ lnRWL + medium + mediumCross)
summary(Fit)
drop1(Fit, test="F")

Fit <- lm(lnMass ~ lnRWL + medium)
summary(Fit)
drop1(Fit, test="F")

Output from xlispstat:

• Regression:
Linear Regression:        Estimate            SE              Prob

Constant                 -2.74044       (0.803583)          0.00085
lnRWL                    2.61456        (0.244668)          0.00000
medium-ind               -5.52714       (2.58019)           0.03395
large-ind                0.682505       (2.92283)           0.81572
medium-ind*lnRWL         1.58078        (0.695402)          0.02457
large-ind*lnRWL          -0.100850      (0.750453)          0.89330

R Squared:               0.983722
Sigma hat:               0.112296
Number of cases:               143
Degrees of freedom:            137

• Step-wise Regression:
Linear Regression:        Estimate            SE              Prob

Constant                 -4.63425       (0.146302)          0.00000
lnRWL                    3.19455        (4.227782E-2)       0.00000

R Squared:               0.975899
Sigma hat:               0.134690
Number of cases:               143
Degrees of freedom:            141

• Backward Step-wise Regression:
Linear Regression:        Estimate            SE              Prob

Constant                 -4.08042       (0.143162)          0.00000
lnRWL                    3.02288        (4.216392E-2)       0.00000
medium-ind*lnRWL         6.088277E-2    (7.979796E-3)       0.00000

R Squared:               0.982977
Sigma hat:               0.113600
Number of cases:               143
Degrees of freedom:            140

Output from R:

• Regression:
Call:
lm(formula = lnMass ~ lnRWL + factor(species) * lnRWL)

Residuals:
Min        1Q    Median        3Q       Max
-0.280596 -0.081156  0.001235  0.082626  0.251145

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)             -2.7404     0.8036  -3.410 0.000853 ***
lnRWL                    2.6146     0.2447  10.686  < 2e-16 ***
factor(species)l         0.6825     2.9228   0.234 0.815715
factor(species)m        -5.5271     2.5802  -2.142 0.033950 *
lnRWL:factor(species)l  -0.1008     0.7505  -0.134 0.893295
lnRWL:factor(species)m   1.5808     0.6954   2.273 0.024570 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1123 on 137 degrees of freedom
Multiple R-squared: 0.9837,     Adjusted R-squared: 0.9831
F-statistic:  1656 on 5 and 137 DF,  p-value: < 2.2e-16

• Step-wise Regression and Backward Step-wise Regression give virtually a dead-heat between the following two models::
Call:
lm(formula = lnMass ~ lnRWL + medium)

Residuals:
Min        1Q    Median        3Q       Max
-0.309325 -0.081741  0.001887  0.090330  0.273934

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.08275    0.14327  -28.50  < 2e-16 ***
lnRWL        3.02359    0.04220   71.66  < 2e-16 ***
medium       0.22861    0.03008    7.60 3.87e-12 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1137 on 140 degrees of freedom
Multiple R-squared: 0.9829,     Adjusted R-squared: 0.9827
F-statistic:  4033 on 2 and 140 DF,  p-value: < 2.2e-16

Call:
lm(formula = lnMass ~ lnRWL + mediumCross)

Residuals:
Min        1Q    Median        3Q       Max
-0.309276 -0.081488  0.001927  0.090330  0.273895

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -4.08042    0.14316  -28.50  < 2e-16 ***
lnRWL        3.02288    0.04216   71.69  < 2e-16 ***
mediumCross  0.06088    0.00798    7.63 3.29e-12 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1136 on 140 degrees of freedom
Multiple R-squared: 0.983,      Adjusted R-squared: 0.9827
F-statistic:  4042 on 2 and 140 DF,  p-value: < 2.2e-16

• Stair-step model: we are playing around with this, finalizing the break-points, standard errors, and using a step-model (rather than the sigmoidal models) for the final plot. For the moment, we have only Katie's model using the sigmoidal transitions, but this will be replace with the stair-step model by Friday, when I've finalized the choice, found the standard errors, and passed those along to Katie, who will make the final figure. In particular, she will have only three symbols, for the small, medium, and large cicada types. See if you can get a legend in the graph, Katie.

Here is the preliminary (I don't believe the final will change much), and I will do the write-up:

I was able to obtain approximate standard error estimates, using our non-linear regression approximations discussed last time:
 baseline 26.588 0.342 step-one 16.038 0.671 step-two 10.96 1.299 center-one 28.372 0.009 width-one 0.008 0.01 center-two 33.664 0.008 width-two 0.007 0.006
. These include the width of the normal cdf sigmoidal functions, but I think that we can simply eliminate those widths (since they are not significantly different from zero) and recalculate. And we won't get much change, although it might be worth trying....