Sunday, August 28, 2011

Comparing two regression slopes by means of an ANCOVA

Regressions are commonly used in biology to determine the causal relationship between two variables. This analysis is most commonly used in morphological studies, where the allometric relationship between two morphological variables is of fundamental interest. Comparing scaling parameters (i.e. slopes) between groups can be used by biologist to assess different growth patterns or the development of different forms or shapes between groups. For examples, the regression between head size and body size may be different between males and females if they grow differently. This difference in allometric growth should manifest itself as a different slope in both regression lines.

Ancova

The analysis of covariance (ANCOVA) is used to compare two or more regression lines by testing the effect of a categorical factor on a dependent variable (y-var) while controlling for the effect of a continuous co-variable (x-var). When we want to compare two or more regression lines, the categorical factor splits the relationship between x-var and y-var into several linear equations, one for each level of the categorical factor.
Regression lines are compared by studying the interaction of the categorical variable (i.e. treatment effect) with the continuous independent variable (x-var). If the interaction is significantly different from zero it means that the effect of the continuous covariate on the response depends on the level of the categorical factor. In other words, the regression lines have different slopes (Right graph on the figure below). A significant treatment effect with no significant interaction shows that the covariate has the same effect for all levels of the categorical factor. However, since the treatment effect is important, the regression lines although parallel have different intercepts. Finally, if the treatment effect is not significant nor its interaction with the covariate (but the coariate is significant), this means there is a single regression line. A reaction norm is used to graphically represent the possible outcomes of an ANCOVA.
ancova-fig-2
For this example we want to determine how body size (snout-vent length) relates to pelvic canal width in both male and female alligators (data from Handbook of Biological Statistics). In this specific case, sex is a categorical factor with two levels (i.e. male and female) while snout-vent length is the regressor (x-var) and pelvic canal width is the response variable (y-var). The ANCOVA will be used to assess if the regression between body size and pelvic width are the comparable between the sexes.

Interpretation

An ANCOVA is able to test for differences in slopes and intercepts among regression lines. Both concepts have different biological interpretations. Differences in intercepts are interpreted as differences in magnitude but not in the rate of change. If we are measuring sizes and regression lines have the same slope but cross the y-axis at different values, lines should be parallel. This means that growth is similar for both lines but one group is simply larger than the other. A difference in slopes is interpreted as differences in the rate of change. In allometric studies, this means that there is a significant change in growth rates among groups.
Slopes should be tested first, by testing for the interaction between the covariate and the factor. If slopes are significantly different between groups, then testing for different intercepts is somewhat inconsequential since it is very likely that the intercepts differ too (unless they both go through zero). Additionally, if the interaction is significant testing for natural effects is meaningless (see The Infamous Type III SS). If the interaction between the covariate and the factor is not significantly different from zero, then we can assume the slopes are similar between equations. In this case, we may proceed to test for differences in intercept values among regression lines.

Performing an ANCOVA

For an ANCOVA our data should have a format very similar to that needed for an Analysis of Variance. We need a categorical factor with two or more levels (i.e. sex factor has two levels: male and female) and at least one independent variable and one dependent or response variable (y-var).
> head(gator)
sex snout pelvic
1 male  1.10   7.62
2 male  1.19   8.20
3 male  1.13   8.00
4 male  1.15   9.60
5 male  0.96   6.50
6 male  1.19   8.17
The preceding code shows the first six lines of the gator object which includes three variables: sex, snout and pelvic, which hold the sex, snout-vent size and the pelvic canal width of alligators, respectively. The sex variable is a factor with two levels, while the other two variables are numeric in their type.
We can do an ANCOVA both with the lm() and aov() commands. For this tutorial, we will use the aov() command due to its simplicity.
> mod1 <- aov(pelvic~snout*sex, data=gator)

> summary(mod1)             Df Sum Sq Mean Sq  F value    Pr(>F)
snout        1 51.871  51.871 134.5392 8.278e-13 ***
sex          1  2.016   2.016   5.2284   0.02921 *
snout:sex    1  0.005   0.005   0.0129   0.91013
Residuals   31 11.952   0.386                   
---


Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
The previous code shows the ANCOVA model, pelvic is modeled as the dependent variable with sex as the factor and snout as the covariate. The summary of the results show a significant effect of snout and sex, but no significant interaction. These results suggest that the slope of the regression between snout-vent length and pelvic width is similar for both males and females.
A second more parsimonious model should be fit without the interaction to test for a significant differences in the slope.
> mod2 <- aov(pelvic~snout+sex, data=gator)
> summary(mod2)
        Df Sum Sq Mean Sq  F value    Pr(>F)
snout        1 51.871  51.871 138.8212 3.547e-13 ***
sex          1  2.016   2.016   5.3948   0.02671 *
Residuals   32 11.957   0.374                   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 
The second model shows that sex has a significant effect on the dependent variable which in this case can be interpreted as a significant difference in ‘intercepts’ between the regression lines of males and females. We can compare mod1 and mod2 with the anova() command to assess if removing the interaction significantly affects the fit of the model:
> anova(mod1,mod2)
Analysis of Variance Table

Model 1: pelvic ~ snout * sex
Model 2: pelvic ~ snout + sex
Res.Df    RSS Df  Sum of Sq      F Pr(>F)
1     31 11.952                        
2     32 11.957 -1 -0.0049928 0.0129 0.9101
The anova() command clearly shows that removing the interaction does not significantly affect the fit of the model (F=0.0129, p=0.91). Therefore, we may conclude that the most parsimonious model is mod2. Biologically we observe that for alligators, body size has a significant and positive effect on pelvic width and the effect is similar for males and females. However, we still don’t know how the slopes change.
At this point we are going to fit linear regressions separately for males and females. In most cases, this should have been performed before the ANCOVA. However, in this example we first tested for differences in the regression lines and once we were certain of the significant effects we proceeded to fit regression lines.
To accomplish this, we are now going to sub-set the data matrix into two sets, one for males and another for females. We can do this with the subset() command or using the extract functions []. We will use both in the following code for didactic purposes:
> machos <- subset(gator, sex=="male")
> hembras <- gator[gator$sex=='female',]
Separate regression lines can also be fitted using the subset option within the lm() command, however we will use separate data frames to simplify the creation of graphs:
> reg1 <- lm(pelvic~snout, data=machos); summary(reg1)
Call:
lm(formula = pelvic ~ snout, data = machos)

Residuals:
 Min       1Q   Median       3Q      Max
-0.85665 -0.40653 -0.08933  0.04518  1.57408

Coefficients:
        Estimate Std. Error t value Pr(>|t|)
(Intercept)   0.4527     0.9697   0.467    0.647
snout         6.5854     0.8625   7.636 6.85e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.7085 on 17 degrees of freedom
Multiple R-squared: 0.7742, Adjusted R-squared: 0.761
F-statistic:  58.3 on 1 and 17 DF,  p-value: 6.846e-07

> reg2 <- lm(pelvic~snout, data=hembras); summary(reg2)
Call:
lm(formula = pelvic ~ snout, data = hembras)

Residuals:
 Min       1Q   Median       3Q      Max
-0.69961 -0.19364 -0.07634  0.04907  1.15098

Coefficients:
        Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.2199     0.9689  -0.227    0.824
snout         6.7471     0.9574   7.047  5.8e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.4941 on 14 degrees of freedom
Multiple R-squared: 0.7801, Adjusted R-squared: 0.7644
F-statistic: 49.67 on 1 and 14 DF,  p-value: 5.797e-06 
The regression lines indicate that males have a higher intercept (a=0.45) than females (a=-0.2199), which means that males are larger. We can now plot both regression lines as follows:
> plot(pelvic~snout, data=gator, type='n')
> points(machos$snout,machos$pelvic, pch=20)
> points(hembras$snout,hembras$pelvic, pch=1)
> abline(reg1, lty=1)
> abline(reg2, lty=2)
> legend("bottomright", c("Male","Female"), lty=c(1,2), pch=c(20,1) )
The resulting plot shows the regression lines for males and females on the same plot.
fig-ancova-1

Advanced

We can fit both regression models with a single call to the lm() command using the nested structure of snout nested within sex (i.e. sex/snout) and removing the single intercept for the model so that separate intercepts are fit for each equation.
> reg.todo <- lm(pelvic~sex/snout - 1, data=gator)
> summary(reg.todo)
Call:
lm(formula = pelvic ~ sex/snout - 1, data = gator)
Residuals:
  Min       1Q   Median       3Q      Max
-0.85665 -0.33099 -0.08933  0.05774  1.57408
Coefficients:
             Estimate Std. Error t value Pr(>|t|)
sexfemale        -0.2199     1.2175  -0.181    0.858
sexmale           0.4527     0.8498   0.533    0.598
sexfemale:snout   6.7471     1.2031   5.608 3.76e-06 ***
sexmale:snout     6.5854     0.7558   8.713 7.73e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.6209 on 31 degrees of freedom
Multiple R-squared: 0.9936,     Adjusted R-squared: 0.9928
F-statistic:  1213 on 4 and 31 DF,  p-value: < 2.2e-16

Conclusion

The results shown in the previous figure are consistent with our previous findings, pelvic width is positively related to snout-vent length. The relationship is linear for both males and females. The regression slope is positive and similar for both males and females (b ≈ 7.07; weighted average), which means that pelvic width grows faster than snout-vent length. Finally, the regression line of males intercepts with the y-axis at a higher value than for females, which means that males are larger.

Sunday, January 16, 2011

Confidence Intervals for Regression plot

I was recently confronted with the problem of plotting 95% confidence intervals on a regression line.  I originally thought this would be a straight forward process in R, but soon came to realize there are some intricacies that need to be tackled before completing a presentable plot.

For this example we will construct a regression plot between island habitat area (log transformed) and species number on each island. First we create vectors with the data:

area <- c(153750,15860,5800,5545,11970,2000,100)

species <- c(36,34,33,33,21,12,3)

Logarithms can be applied during the regression, but this procedure creates a problem later when predicted values are obtained from the model (see below). Therefore it is important to create new vectors with the log-transformed values

larea <- log(area)

lsp <- log(species)

 

We now proceed to calculate the regression between log(area) and log(sp).  Based on Island Biogeography Theory (IBT), the area of the habitat should predict species number on a logarithmic scale.  We perform a regression analysis using the lm() command.  The results of the regression are stored in the mod1 object and printed with the summary() command.:

mod1 <- lm(lsp~larea)

summary(mod1)

Call:
lm(formula = lsp ~ larea)

Residuals:
      1       2       3       4       5       6       7
-0.5609  0.1960  0.5266  0.5428 -0.1850 -0.1034 -0.4161

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) -0.13566    0.77702  -0.175  0.86825  
larea        0.35837    0.08744   4.098  0.00937 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4781 on 5 degrees of freedom
Multiple R-squared: 0.7706,    Adjusted R-squared: 0.7247
F-statistic:  16.8 on 1 and 5 DF,  p-value: 0.00937

A quick review of the results show that area significantly predicts species number, as expected from the IBT.  One would now proceed to determine the validity and quality of the analysis through plotting of residuals and related techniques.  Since we are concerned with plots we will skip these analysis in the current post.

To plot the confidence intervals we need to use the predict() command which is used, as its name implies, to predict y-values based on a particular regression model.  The command is actually a wrapper command which applies to different models (lm, glm) and it will adapt itself to the proper analysis. In our case, since we used a liner model (lm), predict()will invoke the predict.lm() command.  For more information type ?predict.lm in the R prompt. 

To illustrate the use of predict(), type in the following commands:

larea
[1] 11.943083  9.671555  8.665613  8.620652  9.390159  7.600902  4.605170

lsp
[1] 3.583519 3.526361 3.496508 3.496508 3.044522 2.484907 1.098612

predict(mod1)
       1        2        3        4        5        6        7
4.144421 3.330367 2.969864 2.953751 3.229522 2.588300 1.514710

The previous code shows the values for the log-transformed area, the number of species, also log-transformed and the predicted values according to the regression equation: lsp = larea*(0.35837) - 0.13566 .  These means that the predicted number of species for an area of 11.94 is 4.144, on the logarithmic scale.   The values returned by predict() are always located on the regression line.

A simple scatterplot and regression line can be produced with the following commands:

plot(larea, lsp)

abline(mod1)

The predict() command calculates confidence intervals for the predicted values.

 

predict(mod1, interval="confidence")

       fit       lwr      upr
1 4.144421 3.2690777 5.019765
2 3.330367 2.8114190 3.849315
3 2.969864 2.5052945 3.434434
4 2.953751 2.4891850 3.418317
5 3.229522 2.7355129 3.723531
6 2.588300 2.0681003 3.108500
7 1.514710 0.4952322 2.534188
 

The previous code shows how to compute confidence intervals for larea values in the regression model (i.e. mod1).  By adding the option interval= “confidence”, lower (lwr) and upper (upr) confidence intervals are computed alongside fitted values.  By default 95% confidence intervals are calculated, if you require higher confidence the level=0.99 may be used.  We could plot these confidence intervals on top of the previous figure, by storing the results of the predict command on an object and then using the lines() command:

 

a <- predict(mod1, interval="confidence")

lines(larea, a[,2], lty=2)

lines(larea, a[,3], lty=2)

r2

This graph seems a bit “choppy” and this effect is due to having few data points in the regression.  We can solve this problem by predicting a larger number of values and their corresponding confidence intervals.  This should result in smoother confidence interval lines.  To do this we need to create a vector with a large number of values in the range of the original x-variable (i.e. larea).  We then use these new values to calculate predicted values (which we wont use) and their confidence intervals to plot them as curves.

newx <- seq(min(larea), max(larea), 0.1)

a <- predict(mod1, newdata=data.frame(larea=newx), interval="confidence")

 

Now, these two last commands require a detailed explanation.  the first one creates a  sequence starting on larea’s minimum value, ending on its maximum value taking 0.1 steps.  The newx vector contains a total of 74 values.

The second command calculates predicted y-values for newx.  In order to accomplish this we need to tell R to use newx as a new set of data and use it as if it were larea.  This is accomplished by the option: newdata=data.frame(larea=newx)

It is very important, that we use the same name as the x-variable in the model (i.e. larea in mod1), otherwise R will complain about vectors of different size.  Now, the only thing left to do is to redo the plot and add the lines:

plot(larea,lsp)

abline(mod1)

lines(newx,a[,2], lty=3)

lines(newx,a[,3], lty=3)

It is very important to use newx in the lines() command otherwise the length of vectors will not match. 

r3

 

the lines in this final graph are a lot smoother.

Saturday, November 22, 2008

Spatial Statistics. Part I.

Spatial Statistics. Part I.

Ripley's K(r) function

Spatial statistics are becoming an important tool for population ecologists. These techniques allow researchers to answer questions regarding the spatial arrangements of individuals within a population and the causal factors that may be influencing spatial distributions. I will detail some basic spatial statistics using the spatstat library.
library(spatstat)
The first approach to a planar point pattern (ppp) is to describe the size of the plot, the number of individuals (i.e. points) and the density of such occurrences or instances. In spatial statistics density is usually referred to as intensity. We will be using the bei data-set included in the spatstat library. From the help page: «The dataset bei gives the positions of 3605 trees of the species Beilschmiedia pendula (Lauraceae) in a 1000 by 500 metre rectangular sampling region in the tropical rainforest of Barro Colorado Island.» The following code invokes the bei data-set and requests some summary statistics:
> data(bei) > summary(bei)
Planar point pattern: 3604 points Average intensity 0.00721 points per square metre Window: rectangle = [0, 1000] x [0, 500] metres Window area = 5e+05 square metres Unit of length: 1 metre
> plot(bei)
The previous results show a total of 3604 points in a 50 ha plot. The average intensity (i.e. density) is 0.00721 points per square meter or 72.1 trees per hectare. The position of the trees is shown in the following graph produced by the plot(bei) command:


The plot evidently shows that this species --B. pendula -- is located more commonly in certain areas of the plot, while absent in others. This suggests an aggregated spatial distribution, which is commonplace for tropical trees. To assess the spatial distribution of these points, we will use Ripley's K(r) function. This method, also known as Ripley's second moment reduced function, estimates the expected number of random points within a distance r of a randomly chosen point within a plot (Ripley 1976). Ripley's K(r) function is generally transformed as follows:
The L(r) function transforms the theoretically expected value for a random distribution into a horizontal line intersecting the P(0,0) point, thus it is more easily interpreted than the exponential K(r) function. Ripley's K(r) function is produced by the Kest() command in R. Given the large number of points in the bei data-set, calculations may take a while in computers with slow processors. The L(r) transformation is performed on-the-fly by the plot() command.
> a1 <- Kest(bei, correction="isotropic", nlarge=Inf) > plot(a1, sqrt(./pi)-r~r, ylab="L(r)")
The previous code request Ripley's K(r) function using the bie data-set. We specifically request the isotropic correction for edge effects. Spatstat's Kest() function has a restriction for data-sets larger than 3000 points. In order to circumvent this restriction we must include the nlarge=Inf option in the command. Results are stored in the a1 object.

Ripley's L(r) function is shown in the previous graph. The x-axis show the automatically selected radii (r) for which abundances are calculated, while the y-axis shows the L(r) function. The dotted red line is the expected value for a random distribution and the black solid line is the observed count. If observed values lie above the zero line (i.e. random expectation) one should suspect an aggregated distibution. Nevertheless, we need to assess if this deviation is large enough to reject the null hipothesis of a random spatial distribution or CSR (complete spatial randomness). To answer this question we need to create confidence intervals for the null hypothesis using the envelope() command. These calculations require a lot of computer resources given the large number of points in the bei data-set, therefore I reduced the number of simulations to 50 from the default nsim=99:
> sobre <- envelope(bei, Kest, nlarge=Inf, nsim=50)
Generating 50 simulations of CSR ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50.
Done. > sobre
Pointwise critical envelopes for K(r) Obtained from 50 simulations of simulations of CSR Significance level of pointwise Monte Carlo test: 2/51 = 0.0392156862745098 Data: bei Function value object (class 'fv') for the function r -> K(r) Entries: id     label     description --      -----     ----------- r      r      distance argument r obs      obs(r)      function value for data pattern theo      theo(r)      theoretical value for CSR lo      lo(r)      lower pointwise envelope of simulations hi      hi(r)      upper pointwise envelope of simulations -------------------------------------- Default plot formula: . ~ r Recommended range of argument r: [0, 125] Unit of length: 1 metre
> plot(sobre, sqrt(./pi)-r~r, lty=c(1,2,3,3), col=c(1,2,3,3), ylab="L(t)")
The resulting graph is as follows:

Given that the observed count (black solid line) is above the confidence interval (green doted lines) we can conclude that the spatial distribution of B. pendula trees significantly deviates from random expectations.

Conclusion

As expected for a tropical tree (Condit, 2000), Ripley's K(t) shows that B. pendula is spatially aggregated. References:
Condit R, Ashton PS, Baker P, Bunyavejchewin S, Gunatilleke S, Gunatilleke N, Hubbell SP, Foster RB, Itoh A, LaFrankie JV, Lee HS, Losos E, Manokaran N, Sukumar R, Yamakura T (2000) Spatial patterns in the distribution of tropical tree species. Science 288:1414―1418.
Ripley BD (1976) The second-order analysis of stationary point processes. Journal of Applied Probability 13:255-256.
Ripley BD (1977) Modeling spatial patterns. Journal of the Royal Statistical Society, B. 39:172-212.

Saturday, October 20, 2007

The Infamous type III SS

Before I started using R, I was under the impression that there was only one type of sums of squares (SS) that should be used: type III SS. It actually was a bit confusing when some people chose to use other types of SS. I also didn't know that SAS invented the term: type III SS. Once I started using R and came across my first unbalanced factorial design...reality hit me.

First of all, I couldn't get R to deploy my coveted type III SS. I could clearly see that any summary coming from lm() or aov() were clearly type I SS, or additive sums of squares. After some reading, I could get type II SS from different models, but the process seemed a bit to complicated. It had to be simpler!!

I played with the idea of posting in R-help , but I was (again) scared away by the terrible comments from some of the gurus. After searching for endless hours on Google, I came across a bunch of interesting information. I read comments and posts regarding how bad type III SS are, mainly because they allow to test for main effects even in the presence of interaction. Honestly I don't think is all that terrible, but I understand it shouldn't be done. I actually teach it that way to my students: "If an interaction is significant, the main effects are rarely interpretable".

On the other camp, I read proponents suggesting that type III SS are the only ones that allow hypothesis testing, giving that they are order independent. Also very true.

I made up my mind, when someone compared SAS to Micro$oft. From then on, I chose not to use type III SS if possible. Therefore, this entry is to teach how to deal with unbalanced factorials and how to get the appropriate SS and F values from R. And if you're so inclined, I will show you how to get the blasted type III SS's. Most of this information is scattered all over the Internet and I'm not to be credited for any of it. I just collected it for my personal use, and since this Blog serves as my basket case for R methods I easily forget, they ended up here.

Let's begin with an example. Let´s suppose we have a two-way unbalanced ANOVA to study the effect of forest type and species on herbivory. The biological hypothesis suggests that species vary in their susceptibility to herbivory, and the percentage of leaves damaged by herbivory are independent of forest type. We sample three different forest types: riparian, transition and mature forests. In each forest type, we collected samples from four different species and determined the percent of area removed from a randomly selected leaf. Given that species number is not constant on each site, the design is unbalanced. The data is presented in the following table:

Herbivory

Riparian
Transition
Mature
Sp1
42 44 36
13 19 22
33 26 33
21
31 3 25
25 24
Sp2
28 23 34
42 13
34 33 31
36
3 26 28
32 4 16
Sp3
1 29 19
11 9 7
1 6
21 1 9
3
Sp4
24 9 22
2 15
27 12 12
5 16 15
22 7 25
5 12

We read the data and included it in R as a data frame named: herb. Now we can check the number of replicates using the replications() command:

> replications(herbiv~bosque*especie, data=herb)
$bosque
bosque
   maduro    ripario transicion
       20         19         19

$especie
especie
sp1 sp2 sp3 sp4
15  15  12  16

$"bosque:especie"
           especie
bosque       sp1 sp2 sp3 sp4
 maduro       5   6   4   5
 ripario      6   5   3   5
 transicion   4   4   5   6
As we can clearly see, the model is highly unbalanced. We shall perform an analysis of variance in various ways. First we will fit a saturated model, and then vary the order of the factors in the model:
> mod.sat <- aov(herbiv~bosque*especie, data=herb)
> mod.sat.2 <- aov(herbiv~especie*bosque, data=herb)

The summary tables for both models show that SS are calculated sequentially.

> summary(mod.sat)

               Df Sum Sq Mean Sq F value    Pr(>F) 
bosque          2  464.0   232.0  2.4766   0.09516 .
especie         3 2810.8   936.9 10.0017 3.434e-05 ***
bosque:especie  6  530.4    88.4  0.9436   0.47348 
Residuals      46 4309.1    93.7                   

> summary(mod.sat.2)
              Df Sum Sq Mean Sq F value    Pr(>F) 
especie         3 2834.8   944.9 10.0871 3.186e-05 ***
bosque          2  440.0   220.0  2.3486    0.1069 
especie:bosque  6  530.4    88.4  0.9436    0.4735 
Residuals      46 4309.1    93.7                   

Although very close, the SS for the species factor (i.e. especies) is different depending on the order of the factors. If species comes first then (SS = 2834,8), but if species is the second term in the model then SS=2810.8. These SS are all sequential SS´s, known by SAS as Type I SS. In the SAS world, now we would use proc GLM and get type III sums of squares, and test hypothesis regarding the interaction and the main effects. Nonetheless, this approach may lead us down a dangerous path, since it allows us to test for main effects even in the presence of a significant interaction.

The R way The advisable method in R, is to search for the most parsimonious model and then obtain SS by comparing models that differ in the number of parameters (i.e. factors) included. This procedure is automagically done by the drop1() command. This command, as its name implies, drops arguments from the model and compares the original model to the reduced one. It generally begins by removing non-significant interactions. The results given by drop1() command are order-independent, therefore the following code examples will only be performed on mod.sat:

> drop1(mod.sat, test="F")
Single term deletions

Model:
herbiv ~ bosque * especie
              Df Sum of Sq    RSS    AIC F value  Pr(F)
<none>                      4309.1  273.9            
bosque:especie  6 &nbsp;   530.4 4839.5  268.6  0.9436 0.4735
>

The drop1() command used in the previous code, required two parameters or options: the name of the fitted aov() model, and the type of test to perform. We asked R to drop terms from "mod.sat", our saturated model. We also requested F statistics, which compare the original model, with the new model without the term. AIC, the Akaike Information Criterion, measures the goodness of fit of a statistical model taking into account the number of parameters included. The AIC statistic allows us to find the best model that fits the data, with the minimum number of parameters.

One should choose the model with the lowest AIC value. In the previous output, removing the interaction term: bosque:especie; produces a better model with a lower AIC than the saturated formula (268.6 vs 273.9). Therefore, we can conclude that removing the interaction term should benefit our model. The F statistic provided is based on SS calculated by comparing models with and without the interaction term, and hence, can be used to report the non-significance of the interaction.

We now fit a linear model without the interaction term, and request the appropriate sums-of-square with the drop1().

> mod1 <- aov(herbiv~bosque+especie, data=herb)

> drop1(mod1, test="F")

Single term deletions

Model:
herbiv ~ bosque + especie
       Df Sum of Sq    RSS    AIC F value     Pr(F) 
<none>               4839.5  268.6                   
bosque   2     440.0 5279.5  269.6  2.3639    0.1041 
especie  3    2810.8 7650.2  289.2 10.0672 2.461e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The output from the drop1() command shows that removing any of the natural components (i.e. 'bosque' or 'especie') would lower the goodness of fit of the model. This can be seen by inspecting the AIC statistics. We see that by removing <none> of the factors, we get an AIC of 268.6. However, if we remove 'bosque' or 'especie', we get a higher AIC value, thus suggesting that we have the most parsimonious model.

F-statistics are computed by comparing the original model, with one where the factor is removed. We can see that 'bosque' is not significant, while 'especie' is highly significant (F=10.07; df=2,46; p<0.001). As I precieve it, these are all type III SS, but I could be wrong. We can clearly see that they are order independent, by fitting a model where 'especie' comes first:

> drop1(aov(herbiv~especie+bosque, herb), test="F") Single term deletions

Model:
herbiv ~ especie + bosque
       Df Sum of Sq    RSS    AIC F value     Pr(F) 
<none>               4839.5  268.6                   
especie  3    2810.8 7650.2  289.2 10.0672 2.461e-05 ***
bosque   2     440.0 5279.5  269.6  2.3639    0.1041 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The results do not deviate from the previous analysis, thus confirming that this type of analysis is order independent. My only grievance with this type of analysis, is that the ANOVA table has to be computed by hand. I still haven't figured out if this can be done automatically.

Type III SS from R

If you want to get type III sums-of-square anyway (maybe due to extraordinary circumstances: your boss demands them!) you can easily get them from R. As Dr. Venables says: you just have to know where to look for them.

I will not go into detail on how this works, mainly, because I don't know how it works. To get type III-SS, you have to do the following:

1. Change the default contrast matrix used by R, to one which produces orthogonal contrasts such as contr.helmert or contr.sum:

> options(contrasts=c("contr.helmert", "contr.poly"))

2. Fit a new anova model (saturated), which uses the new contrast matrix we selected in step 1.

> mod.III <- aov(herbiv~bosque*especie, data=herb)

3. Then use the drop1() command, choosing to remove all objects using the formula shorthand notation .~. :

> drop1(mod.III, .~., test="F") Single term deletions

Model:
herbiv ~ bosque * especie 

Df Sum of Sq RSS AIC F value Pr(F) <none> 4309.1 273.9 bosque 2 436.3 4745.4 275.5 2.3289 0.1088 especie 3 2753.8 7062.8 296.5 9.7989 4.108e-05 *** bosque:especie 6 530.4 4839.5 268.6 0.9436 0.4735 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

And voila!...You get type III sums-of-square, which should make any SAS user happy. The Car package, has built in functions to get type-III SS, but I've read they produce funny output sometimes. I've never tested this, since I don't like installing many libraries to do things you can get from the base stats module.

Conclusion

In my opinion the type III sums-of-square issue, is a bit technical for the average user. Although some suggest that the average user shouldn't be doing statistics, it is commonplace for many students and some researchers to have to deal with anova on their own, with very little statistics background. I think R does the correct thing by not supplying the infamous SS's easily, and forcing the user to think about what he/she really wants. Nonetheless, this discussion should be part of R's help files, and the correct procedure should be also easier to find. Finally, the drop1() method should provide a properly formatted ANOVA table, but again, I may just not know how to do it...and the correct procedure is hidden somewhere within R's mysterious innards.

Saturday, October 06, 2007

Contrasts in R

One of the most neglected topics in Biostatistics is the calculation of contrasts in ANOVA. Most researchers are content with simply calculating an ANOVA table and stating that differences among groups are statistically significant. Some even present box-plots or any other graphic that show where the differences lie. However, few researchers actually test the differences they are looking for.

An ANOVA is designed to test a statistical hypothesis. It looks at the means of each treatment or level, and determines if all the means are equal or not. Nevertheless, the biological hypotheses introduced by the researchers need to be specifically addressed with contrasts. One can generalize and state that while ANOVA analyzes the statistical hypothesis, contrasts look into the biological hypothesis.

The following entry will show how to create a contrast in R. I will not go into detail regarding the theory behind contrasts. I suggest you read a good book on Experimental Design.

Let’s begin by using a data set included in R. We will use the InsectSprays data frame. This experiment shows the number of insects killed (variable count) by using six different insecticides (A:F). The data is loaded by using:

> data(InsectSprays)

Once the data has been invoked, we can start working on it. The first thing we want to do is to determine if the model is balanced. We can use the summary() command to assess some basic information on the data.frame:

> summary(InsectSprays)

count spray

Min. : 0.00 A:12

1st Qu.: 3.00 B:12

Median : 7.00 C:12

Mean : 9.50 D:12

3rd Qu.:14.25 E:12

Max. :26.00 F:12

The previous output shows that we have a balanced design, with 12 replicates per group. Since this is a dataframe the alphanumeric variable, spray, is immediately recognized by R as a factor.

Let’s imagine there are three main questions. First, we want to determine if there are any differences between the spray treatments. Secondly, the researcher wants to know if the first three sprays differ from the latter three. Third, there is strong indication that sprays: A, B, and F; should have a stronger effect than the remaining treatments. Since contrasts are pre-planned or a priori comparisons, we should start by creating the contrasts matrix. We need to create a vector of coefficients for each comparison, and then bind them in a matrix:

> c1 <- c(1,1,1,-1,-1,-1)

> c2 <- c(1,1,-1,-1,-1,1)

> mat <- cbind(c1,c2)

Secondly, we need to specify that the contrast matrix mat should be used to compute SS during anova calculations. We achieve this by assigning the contrasts matrix to the factor sprays using the contrasts() command:

> contrasts(InsectSprays$spray) <- mat

It should be noted, that once this assignment is implemented, the aov() command will calculate SS for the contrasts established in each column of mat. If you wish to change the contrasts, then a new contrasts matrix should be created and assigned to the factor.

We now perform the analysis of variance, and request a summary table. The anova table is split to include the contrasts:

> model1 <- aov(count ~ spray, data = InsectSprays)

> summary(model1, split=list(spray=list("First 3 vs other"=1, "ABF vs CDE"=2)))

Df Sum Sq Mean Sq F value Pr(>F)
spray5

2668.83

533.77

34.702

<2e-16

***
spray: First 3 vs other 1

93.39

93.39

6.0716

0.01635

*
spray: ABF vs CDE 1

2558.67

2558.67

166.3495

< 2e-16

***
Residuals 66

1015.17

15.38

---

Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

Let’s analyze the previous code. First we conduct an analysis of variance with the aov() command. The results are stored in the model1 object. Immediately following, we request a summary table with the summary() command. The summary command includes the option: split. The split option provides a list of factors where contrasts are stored. Within each factor, we also provide a list which includes the names of the contrasts (i.e. each column) stored in the contrasts matrix columns (i.e. mat).

The ANOVA table shows a significant effect of the spray varieties on insect moratility. The contrasts suggest significant differences between the first three and the latter three spray types. Finally, we see that most of the treatment SS are explained by the comparison of the ABF and CDE groups. A box-plot clearly shows that the main differences lie between the groups compared by the second contrasts. The comparisons among groups are better observed using a boxplot:

> boxplot(count ~ spray, data = InsectSprays, + xlab = "Type of spray", ylab = "Insect count", + main = "InsectSprays data", col = "skyblue")

Saturday, September 15, 2007

Neighbor Joining Tree with Ape

Today I used R to create a neighbor joining phenogram, using the ape library. The phenogram will be used to visualize a species genetic differences between six different locations in Costa Rica. Genetic distances were calculated from microsatellite data using GenAlEx (GenAlEx), because I haven't figured out how to calculate Nei's genetic dissimilarities with R. This is the procedure I used:

First I imported the genetic distance matrix from Exc..l. The matrix used was symmetric, meaning that the same distances were mirrored above and below the diagonal. I copied the matrix into the clipboard from Exc..l using Ctrl-C. The distance matrix included row and coloumn names, the six different locations. The matrix was imported in R by assigning it to an object named m:

> m <- as.matrix(read.table("clipboard", head=T, row.names=1))

The previous command imports the data stored in the clipboard and transforms the data frame into a matrix format. The options of the read.table() command: head=T and row.names=1, tell R that the first line of the matrix is a header and that row names are stored in the first column, respectively. After checking the matrix was imported correctly, I proceeded to load the ape library.

> library(ape)

The ape (analysis in phylogenetics and evolution) contains various methods for the analysis of genetic and evolutionary data. Ape also provides commands designed to calculate distances in DNA sequences. I will tackle those in another post. Now back to my tree. After importing the distance matrix, and including the ape library, I proceeded to create the phylogenetic tree with the nj() command:

> arbol <- nj(as.dist(m), "unrooted") > plot(arbol)

The "unrooted" parameter provided to the nj() command produces, as expected, an unrooted tree. The resulting tree topography was stored in the 'arbol' object. The plot command produced:

We can observe how 'Puriscal' is located between two well demarcated groups, one including 'Guapiles' and 'Guatuso' and the other three populations in the second group.


Friday, September 29, 2006

Mantel Partial Regression

To perform a partial mantel regression on three different distance matrices: > library(vegan) > mantel.partial (xdis, ydis, zdis, + method = "pearson", permutations = 1000) Example. Matrix 1: Genetic distance, Nei's D. Matrix 2: Geographic distance Matrix 3: Regional correspondance (i.e. binary matrix, 1 if two populations belon to the same region, zero otherwise) This will allow to test if a correlation between geographic distance and genetic differentiation exists, when regional correspondance is taken into account.