Welcome to the Analyzing Continuous Data Unit! In this series of lessons, we are looking at how to do various common statistical tests in R, leading to more complex data and modelling approaches.

In the previous Unit, we have looked at solely categorical data (Testing Ratios) and continuous data from categories (Testing Populations).

In this lesson, we will look at methods to test whether two or more samples of continuous data are positively or negatively associated. We will also consider having multiple predictor variables. Our fundamental question is:

Is There An Association Between Two (or More) Variables?

Categorical Variables

For categorical variables, we can use a Chi-Squared test of association. See the Testing Ratios lesson for more details.

Two Continuous Variables

Consider that we have two (2) continuous variables. A first question might be: ‘Do these samples come from the same distribution?’ we can use a Kolmogorov-Smirnov Test to answer this.

The function ks.test() performs a two-sample test of the null hypothesis that x and y were drawn from the same continuous distribution.

Let’s generate some random data that we can use to test this function. Use the function rnorm() to generate 50 values drawn randomly from a normal distribution. rnorm() requires the number of samples (n =). You can also change the mean and standard deviation. In this case, just generate 50 values with the defaults (mean of 0 and sd of 1). Assign this to x.

x <- rnorm(50)

Now, generate a vector of 50 values from a uniform distribution using runif(), and assign this to y.

y <- runif(50)

Feel free to use hist() at any point if you’re curious how a normal and uniform distribution vary.

The ks.test() function takes x = and y =. Put our two new objects in there and test if they are from the same distribution.

ks.test(x, y)
## 
##  Two-sample Kolmogorov-Smirnov test
## 
## data:  x and y
## D = 0.42, p-value = 0.000246
## alternative hypothesis: two-sided

The p-value less than 0.001 suggests that x and y are indeed drawn from different distributions (as we suspected … ).

Correlation

Whether we run correlation or a regression depends on how we think the two variables are related. If we think that there is no causal relationship between the two, then we would test for a correlation. If we think that there is a causal relationship between the two, then we would run a regressin.

As they saying goes ‘correlation does not imply causation’.

The function cor() returns the correlation coefficient of two variables. It requires an x = and a y =, and a method =. Pearson’s product moment correlation coefficient (method = 'pearson') is the parametric version used for normal data (and the default). Kendall’s tau (method = 'kendall') or Spearman’s rho (method = 'spearman') are used for non-parametric data.

The function cor.test() tests for association—correlation—between paired samples, using one of Pearson’s product moment correlation coefficient, Kendall’s tau, or Spearman’s rho, as above. The three methods each estimate the association between paired samples and compute a test of the value being zero (indicating no association).

In the New Haven Road Race data, it would seem sensible that Net time and Pace are correlated.

The race data are loaded with the lesson as race. Calculate the correlation coefficient of Net time and Pace.

cor(x = race$Nettime_mins, y = race$Pace_mins)
## [1] 0.9953277

Now test if this correlation coefficient is greater than 0 with cor.test().

cor.test(x = race$Nettime_mins, y = race$Pace_mins)
## 
##  Pearson's product-moment correlation
## 
## data:  race$Nettime_mins and race$Pace_mins
## t = 529.66, df = 2640, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.9949582 0.9956703
## sample estimates:
##       cor 
## 0.9953277

As usual, you can assign this to an object, and extract the various parts of the object, such as the coefficient, p-value, etc.

Simple Linear Regression

Ok, now we are coming to one of the most commonly used statistical models. If we think that one variable/s is driving variation in the other, we should use regression rather than correlation.

The function lm() is used to develop linear models. At its simplest, it takes one argument: formula =. This argument is the first one, and so folks rarely even write out the formula = part.

However, this formula is how models (i.e., statistical relationships between variables) are specified, symbolically. A typical model has the form response ~ terms, where response is the response (or dependent) variable and terms is (a series of) term/s which specifies a linear predictor (or independent) variable.

The response variable is fitted as a function of the predictor variable. You have used this already with the tilde symbol (~), which signifies ‘as a function of’, and separates the response and predictor variables. Adding further predictor variables to the right hand side is also possible (see Formulae).

For now, we will carry out a simple regression of a single predictor and response. This model estimates values for the three elements of the equation: y ~ beta0 + beta1 * x + sigma.

In other words: y ~ intercept + slope * x + sd of the error. We will come back to this later.

We will illustrate the use of lm() using the sparrow data, loaded with this lesson (even though we know there is actually no causality here).

First, as always, look at the data. Display the head of the sparrow data, sparrow.

head(sparrow)
##   Species    Sex Wingcrd Tarsus Head Culmen Nalospi   Wt Observer Age
## 1    SSTS   Male    58.0   21.7 32.7   13.9    10.2 20.3        2   0
## 2    SSTS Female    56.5   21.1 31.4   12.2    10.1 17.4        2   0
## 3    SSTS   Male    59.0   21.0 33.3   13.8    10.0 21.0        2   0
## 4    SSTS   Male    59.0   21.3 32.5   13.2     9.9 21.0        2   0
## 5    SSTS   Male    57.0   21.0 32.5   13.8     9.9 19.8        2   0
## 6    SSTS Female    57.0   20.7 32.5   13.3     9.9 17.5        2   0

Second, let’s plot the data that we want to model. Plot Tarsus as a function of Wingcrd, making sure that you use a formula and the data = argument.

plot(Tarsus ~ Wingcrd, data = sparrow)

It looks like there is a positive relationship between the two. Ok, now we are ready to run the actual model. Call the output m0, and use lm() to model Tarsus as a function of Wingcrd. Notice that you can use almost the same code as you used to plot the data.

m0 <- lm(Tarsus ~ Wingcrd, data = sparrow)

Ok, so our model is stored in the object m0. We can use several functions to extract all or part of this object. As usual, we can use str() to look at everything contained in the model object m0. Try that.

str(m0)
## List of 12
##  $ coefficients : Named num [1:2] 8.374 0.226
##   ..- attr(*, "names")= chr [1:2] "(Intercept)" "Wingcrd"
##  $ residuals    : Named num [1:979] 0.1935 -0.0669 -0.733 -0.433 -0.2801 ...
##   ..- attr(*, "names")= chr [1:979] "1" "2" "3" "4" ...
##  $ effects      : Named num [1:979] -671.968 16.218 -0.74 -0.44 -0.285 ...
##   ..- attr(*, "names")= chr [1:979] "(Intercept)" "Wingcrd" "" "" ...
##  $ rank         : int 2
##  $ fitted.values: Named num [1:979] 21.5 21.2 21.7 21.7 21.3 ...
##   ..- attr(*, "names")= chr [1:979] "1" "2" "3" "4" ...
##  $ assign       : int [1:2] 0 1
##  $ qr           :List of 5
##   ..$ qr   : num [1:979, 1:2] -31.289 0.032 0.032 0.032 0.032 ...
##   .. ..- attr(*, "dimnames")=List of 2
##   .. .. ..$ : chr [1:979] "1" "2" "3" "4" ...
##   .. .. ..$ : chr [1:2] "(Intercept)" "Wingcrd"
##   .. ..- attr(*, "assign")= int [1:2] 0 1
##   ..$ qraux: num [1:2] 1.03 1.02
##   ..$ pivot: int [1:2] 1 2
##   ..$ tol  : num 1e-07
##   ..$ rank : int 2
##   ..- attr(*, "class")= chr "qr"
##  $ df.residual  : int 977
##  $ xlevels      : Named list()
##  $ call         : language lm(formula = Tarsus ~ Wingcrd, data = sparrow)
##  $ terms        :Classes 'terms', 'formula'  language Tarsus ~ Wingcrd
##   .. ..- attr(*, "variables")= language list(Tarsus, Wingcrd)
##   .. ..- attr(*, "factors")= int [1:2, 1] 0 1
##   .. .. ..- attr(*, "dimnames")=List of 2
##   .. .. .. ..$ : chr [1:2] "Tarsus" "Wingcrd"
##   .. .. .. ..$ : chr "Wingcrd"
##   .. ..- attr(*, "term.labels")= chr "Wingcrd"
##   .. ..- attr(*, "order")= int 1
##   .. ..- attr(*, "intercept")= int 1
##   .. ..- attr(*, "response")= int 1
##   .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv> 
##   .. ..- attr(*, "predvars")= language list(Tarsus, Wingcrd)
##   .. ..- attr(*, "dataClasses")= Named chr [1:2] "numeric" "numeric"
##   .. .. ..- attr(*, "names")= chr [1:2] "Tarsus" "Wingcrd"
##  $ model        :'data.frame':   979 obs. of  2 variables:
##   ..$ Tarsus : num [1:979] 21.7 21.1 21 21.3 21 20.7 22 20.8 20.1 22.2 ...
##   ..$ Wingcrd: num [1:979] 58 56.5 59 59 57 57 57 57 53.5 56.5 ...
##   ..- attr(*, "terms")=Classes 'terms', 'formula'  language Tarsus ~ Wingcrd
##   .. .. ..- attr(*, "variables")= language list(Tarsus, Wingcrd)
##   .. .. ..- attr(*, "factors")= int [1:2, 1] 0 1
##   .. .. .. ..- attr(*, "dimnames")=List of 2
##   .. .. .. .. ..$ : chr [1:2] "Tarsus" "Wingcrd"
##   .. .. .. .. ..$ : chr "Wingcrd"
##   .. .. ..- attr(*, "term.labels")= chr "Wingcrd"
##   .. .. ..- attr(*, "order")= int 1
##   .. .. ..- attr(*, "intercept")= int 1
##   .. .. ..- attr(*, "response")= int 1
##   .. .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv> 
##   .. .. ..- attr(*, "predvars")= language list(Tarsus, Wingcrd)
##   .. .. ..- attr(*, "dataClasses")= Named chr [1:2] "numeric" "numeric"
##   .. .. .. ..- attr(*, "names")= chr [1:2] "Tarsus" "Wingcrd"
##  - attr(*, "class")= chr "lm"

There is a lot in there! Given that it is a list, we can access any part of it as we would any other list. For example, we could use m0$coefficients to pull out the model coefficients. Try that.

m0$coefficients
## (Intercept)     Wingcrd 
##   8.3738466   0.2264258

There are also a number of generic functions that are built to work on any model object. One of these generic functions is coef(), which we can use to pull out the coefficient estimates of the model. Try putting m0 in a call to coef().

coef(m0)
## (Intercept)     Wingcrd 
##   8.3738466   0.2264258

Both m0$coefficients and coef(m0) return a (identical) named vector: The names of each coefficient estimate are given for each element of the vector. The intercept is the intercept, i.e., when x = 0, at what point the regression line crosses the y-axis. The slope of the regression line is given by the element named for the predictor variable in the model (in this case Wingcrd). The slope means for a unit change in x, what the change in y is. In this case, a 1-unit change in Wingcrd leads to an increase of 0.23 in Tarsus.

Let’s look at a more traditional output of the statistical model. We can use summary() to return that. Do so.

summary(m0)
## 
## Call:
## lm(formula = Tarsus ~ Wingcrd, data = sparrow)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.06691 -0.53297 -0.03297  0.42273  3.04060 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  8.37385    0.61834   13.54   <2e-16 ***
## Wingcrd      0.22643    0.01068   21.21   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7648 on 977 degrees of freedom
## Multiple R-squared:  0.3152, Adjusted R-squared:  0.3145 
## F-statistic: 449.7 on 1 and 977 DF,  p-value: < 2.2e-16

Here we see several parts to the model output. First, the model itself (Call), then a summary of the residuals (Residuals), then details of the coefficients (Coefficients), their values, standard errors, t values, and p values. The p-values correspond to specific tests. First, the test is if the intercept is significantly different from 0. We are not too often interested in this value or significance. What we are usually more interested in is if there is indeed a positive or negative relationship between the two variables in the model. The p-value in this case tells us if the slope is significantly different from 0. In this case it is, and so we infer that as Wingcrd increases, so does Tarsus.

Below this, we also get some information on the remaining unexplained variation, R2 values, and F tests.

The other useful parts of the model that you may want to look at, at least for model-checking, are the residuals, using either m0$residuals or the generic function residuals(). Try either of those.

residuals(m0)
##            1            2            3            4            5 
##  0.193455497 -0.066905764 -0.732970329 -0.432970329 -0.280118677 
##            6            7            8            9           10 
## -0.580118677  0.719881323 -0.480118677 -0.387628286  1.033094236 
##           11           12           13           14           15 
##  0.785945888 -0.127267025  0.319881323 -0.559396155  0.380242584 
##           16           17           18           19           20 
## -0.527267025 -0.732970329 -0.566905764 -0.993331590  0.306668410 
##           21           22           23           24           25 
## -0.559396155 -0.432970329 -0.706544503  0.372732975 -1.172609068 
##           26           27           28           29           30 
##  0.280242584 -0.459396155 -1.119757416  0.193455497  0.153816758 
##           31           32           33           34           35 
## -1.132970329 -0.132970329 -0.480118677  0.333094236  0.146307149 
##           36           37           38           39           40 
## -0.166905764 -0.680118677  0.093455497 -0.593331590  0.819881323 
##           41           42           43           44           45 
##  0.193455497 -0.140479938 -0.732970329 -1.659396155  0.340603845 
##           46           47           48           49           50 
## -0.132970329  0.346307149 -0.066905764 -0.580118677  0.167029671 
##           51           52           53           54           55 
## -0.806544503  0.572732975 -0.232970329 -0.206544503 -1.132970329 
##           56           57           58           59           60 
## -0.866905764  0.314178019 -0.559396155 -1.132970329  0.167029671 
##           61           62           63           64           65 
## -0.093331590 -0.732970329 -0.346183242 -0.959396155 -0.519757416 
##           66           67           68           69           70 
##  0.693455497 -0.285821981  0.440603845  0.367029671 -0.872609068 
##           71           72           73           74           75 
## -0.066905764  0.419881323 -1.100841199 -0.132970329 -0.100841199 
##           76           77           78           79           80 
## -0.600841199  0.706668410 -0.332970329 -0.332970329 -0.246183242 
##           81           82           83           84           85 
## -0.232970329 -0.332970329 -0.272609068 -0.032970329  0.219881323 
##           86           87           88           89           90 
## -0.819757416  0.333094236 -0.885821981  0.367029671  0.293455497 
##           91           92           93           94           95 
##  0.140603845  0.133094236 -1.019757416 -0.059396155 -0.006544503 
##           96           97           98           99          100 
## -0.132970329 -0.859396155  0.199158801 -1.027267025 -0.300841199 
##          101          102          103          104          105 
## -1.453692851 -0.053692851 -0.700841199 -0.732970329  0.072732975 
##          106          107          108          109          110 
## -0.353692851 -1.006544503  0.125584627 -0.527267025 -1.040479938 
##          111          112          113          114          115 
##  0.300965106  0.993455497 -0.859396155 -1.185821981 -0.159396155 
##          116          117          118          119          120 
##  0.172732975 -0.740479938  0.267029671  0.472732975  0.240603845 
##          121          122          123          124          125 
## -0.512247807 -0.759396155 -0.606544503 -0.140479938 -1.180118677 
##          126          127          128          129          130 
##  0.806668410  0.506668410 -0.193331590 -0.506544503 -0.172609068 
##          131          132          133          134          135 
##  0.080242584 -0.606544503  0.233094236 -0.559396155  0.019881323 
##          136          137          138          139          140 
## -0.685821981  0.006668410  0.419881323  0.193455497 -0.559396155 
##          141          142          143          144          145 
## -0.753692851  0.867029671 -1.327267025 -0.072609068 -1.206544503 
##          146          147          148          149          150 
##  0.846307149  0.799158801  0.072732975  0.680242584 -0.059396155 
##          151          152          153          154          155 
## -0.659396155 -0.819757416 -0.306544503 -1.246183242 -0.532970329 
##          156          157          158          159          160 
##  0.346307149 -1.632970329  0.019881323 -1.119757416  0.240603845 
##          161          162          163          164          165 
##  0.167029671 -0.132970329 -0.532970329 -0.959396155 -0.885821981 
##          166          167          168          169          170 
## -0.732970329 -1.232970329 -0.200841199 -0.200841199 -0.759396155 
##          171          172          173          174          175 
## -0.932970329  0.067029671 -0.053692851 -0.046183242 -0.127267025 
##          176          177          178          179          180 
## -1.380118677 -0.053692851 -0.959396155  0.299158801 -0.400841199 
##          181          182          183          184          185 
## -0.853692851  0.167029671 -0.040479938 -0.614054112 -0.706544503 
##          186          187          188          189          190 
##  0.019881323 -0.532970329  0.072732975 -0.580118677 -0.353692851 
##          191          192          193          194          195 
## -0.059396155 -0.119757416 -0.459396155 -0.185821981 -0.385821981 
##          196          197          198          199          200 
##  0.546307149 -0.885821981  0.267029671  0.467029671 -0.599034894 
##          201          202          203          204          205 
## -0.019757416 -0.232970329 -0.359396155  0.659520062 -1.166905764 
##          206          207          208          209          210 
##  0.672732975 -0.600841199  0.846307149  0.872732975 -0.053692851 
##          211          212          213          214          215 
## -0.340479938 -0.080118677 -0.253692851  1.093455497 -0.432970329 
##          216          217          218          219          220 
## -0.346183242 -0.453692851 -0.059396155  0.167029671  0.267029671 
##          221          222          223          224          225 
##  0.053816758 -0.399034894 -0.232970329 -0.606544503 -1.132970329 
##          226          227          228          229          230 
## -0.614054112 -0.727267025 -0.580118677 -1.053692851  0.953816758 
##          231          232          233          234          235 
##  0.072732975 -2.066905764  0.393455497 -0.606544503  0.972732975 
##          236          237          238          239          240 
## -1.059396155 -0.080118677  0.540603845 -0.432970329  0.399158801 
##          241          242          243          244          245 
## -0.840479938 -0.885821981  0.246307149 -1.332970329 -1.080118677 
##          246          247          248          249          250 
## -0.280118677 -0.053692851 -0.480118677 -0.453692851  0.146307149 
##          251          252          253          254          255 
## -1.346183242 -0.606544503 -1.580118677  0.533094236 -0.906544503 
##          256          257          258          259          260 
## -0.340479938 -0.500841199 -0.606544503  0.293455497 -1.227267025 
##          261          262          263          264          265 
## -1.832970329  0.085945888  1.953816758  0.308474715  1.240603845 
##          266          267          268          269          270 
##  0.074539280  0.680242584 -0.238673633  1.548113454  1.400965106 
##          271          272          273          274          275 
##  0.655623063  0.214178019  1.367029671  1.567029671  0.361326367 
##          276          277          278          279          280 
##  1.667029671  1.514178019  1.600965106  0.087752193 -0.178312372 
##          281          282          283          284          285 
## -0.138673633  1.719881323  0.114178019  0.019881323  0.061326367 
##          286          287          288          289          290 
##  1.219881323  2.253816758 -0.917951111  1.034900541  2.487752193 
##          291          292          293          294          295 
##  2.140603845 -1.359396155  0.314178019  2.433094236  0.800965106 
##          296          297          298          299          300 
##  1.140603845  2.114178019  0.053816758  0.225584627  0.080242584 
##          301          302          303          304          305 
##  0.493455497 -0.559396155 -0.459396155 -0.419757416 -0.006544503 
##          306          307          308          309          310 
## -0.232970329 -0.180118677  0.519881323  0.067029671  0.033094236 
##          311          312          313          314          315 
## -0.659396155 -0.166905764 -0.359396155  1.246307149  0.393455497 
##          316          317          318          319          320 
##  0.846307149  0.599158801  0.959520062  0.406668410  0.659520062 
##          321          322          323          324          325 
## -0.906544503  0.293455497 -0.032970329  0.653816758  0.546307149 
##          326          327          328          329          330 
## -0.246183242  0.219881323 -0.480118677  0.093455497  0.067029671 
##          331          332          333          334          335 
## -1.046183242  0.159520062 -0.146183242 -0.180118677 -0.127267025 
##          336          337          338          339          340 
##  0.272732975 -0.506544503 -0.080118677 -1.159396155  0.993455497 
##          341          342          343          344          345 
##  0.519881323  0.419881323 -0.232970329  0.367029671 -0.506544503 
##          346          347          348          349          350 
## -0.046183242  0.333094236 -0.006544503  0.672732975 -0.719757416 
##          351          352          353          354          355 
## -0.346183242  0.467029671  0.040603845  0.067029671  0.499158801 
##          356          357          358          359          360 
##  0.046307149 -0.080118677  0.519881323  0.106668410  0.372732975 
##          361          362          363          364          365 
##  0.733094236  0.293455497  0.419881323 -1.372609068 -0.346183242 
##          366          367          368          369          370 
##  0.572732975  0.067029671  0.140603845 -0.485821981 -1.032970329 
##          371          372          373          374          375 
##  0.840603845  1.119881323  1.019881323 -0.632970329  0.299158801 
##          376          377          378          379          380 
## -0.780118677  0.167029671 -0.232970329 -0.840479938 -0.280118677 
##          381          382          383          384          385 
##  1.472732975  0.267029671  0.193455497  0.319881323  0.659520062 
##          386          387          388          389          390 
##  0.293455497  0.293455497  0.493455497  0.093455497 -0.553692851 
##          391          392          393          394          395 
## -0.259396155  0.067029671 -0.006544503 -0.646183242  0.859520062 
##          396          397          398          399          400 
##  0.480242584  0.646307149  0.246307149  0.067029671  0.459520062 
##          401          402          403          404          405 
##  1.572732975  0.080242584  0.080242584 -0.532970329  0.172732975 
##          406          407          408          409          410 
## -0.259396155  0.453816758  0.493455497  0.280242584  0.846307149 
##          411          412          413          414          415 
##  0.246307149 -0.180118677 -0.032970329 -0.206544503  0.372732975 
##          416          417          418          419          420 
##  0.199158801  0.872732975  0.625584627  0.772732975 -0.206544503 
##          421          422          423          424          425 
##  0.353816758  0.772732975  0.999158801  0.485945888 -0.227267025 
##          426          427          428          429          430 
## -1.006544503  0.693455497  0.346307149 -0.693331590  0.172732975 
##          431          432          433          434          435 
## -0.706544503 -0.206544503 -0.946183242  0.846307149 -0.232970329 
##          436          437          438          439          440 
## -0.032970329 -0.732970329  1.085945888  0.485945888  0.219881323 
##          441          442          443          444          445 
## -1.632970329 -0.153692851 -0.114054112  1.538797540 -0.419757416 
##          446          447          448          449          450 
## -0.780118677  0.246307149 -0.127267025  0.846307149  1.099158801 
##          451          452          453          454          455 
##  0.233094236 -0.106544503  1.372732975 -0.959396155 -1.046183242 
##          456          457          458          459          460 
##  0.819881323  0.753816758  1.093455497 -0.206544503  0.699158801 
##          461          462          463          464          465 
## -0.346183242  0.393455497 -1.080118677  1.146307149  0.506668410 
##          466          467          468          469          470 
##  0.093455497  0.799158801 -0.059396155 -0.132970329 -0.432970329 
##          471          472          473          474          475 
## -0.419757416 -0.019757416 -0.885821981 -0.832970329 -1.519757416 
##          476          477          478          479          480 
##  0.119881323  0.119881323 -0.385821981 -0.593331590 -0.066905764 
##          481          482          483          484          485 
##  0.846307149 -0.246183242 -1.006544503  0.519881323  0.080242584 
##          486          487          488          489          490 
##  0.619881323  0.093455497  0.180242584 -0.253692851 -0.327267025 
##          491          492          493          494          495 
##  1.499158801 -1.032970329  0.672732975 -0.032970329 -0.027267025 
##          496          497          498          499          500 
## -0.580118677  0.559520062  0.306668410  0.119881323  0.619881323 
##          501          502          503          504          505 
##  0.925584627  0.167029671  1.085945888  0.367029671  0.819881323 
##          506          507          508          509          510 
##  0.593455497 -0.932970329  0.006668410  0.772732975  0.499158801 
##          511          512          513          514          515 
## -0.259396155  0.180242584  0.561326367  1.506668410  1.093455497 
##          516          517          518          519          520 
##  1.767029671  2.180242584  1.285945888  0.334900541  2.119881323 
##          521          522          523          524          525 
##  2.114178019  2.340603845  1.987752193 -0.585821981  2.327390932 
##          526          527          528          529          530 
##  2.161326367  0.648113454  1.167029671  1.314178019  1.187752193 
##          531          532          533          534          535 
##  0.748113454  1.119881323  0.519881323  1.014178019  0.814178019 
##          536          537          538          539          540 
##  0.387752193  1.253816758  2.219881323  0.067029671 -0.732970329 
##          541          542          543          544          545 
## -0.406544503  0.193455497  0.946307149 -0.432970329 -0.080118677 
##          546          547          548          549          550 
##  0.072732975  0.419881323 -0.659396155 -0.132970329 -0.506544503 
##          551          552          553          554          555 
## -0.200841199 -0.380118677  0.772732975  0.072732975 -0.559396155 
##          556          557          558          559          560 
##  0.119881323  0.699158801 -0.546183242 -0.553692851 -0.353692851 
##          561          562          563          564          565 
##  0.272732975 -0.280118677 -0.880118677 -0.140479938  0.093455497 
##          566          567          568          569          570 
## -0.485821981 -0.332970329 -0.966905764  0.472732975  0.172732975 
##          571          572          573          574          575 
## -1.206544503 -0.206544503  0.472732975 -1.246183242  0.093455497 
##          576          577          578          579          580 
##  0.012371714  0.067029671  0.093455497 -0.546183242 -1.332970329 
##          581          582          583          584          585 
##  0.485945888 -0.653692851  0.046307149 -0.353692851 -0.453692851 
##          586          587          588          589          590 
## -0.380118677 -0.706544503 -0.159396155  0.919881323  0.119881323 
##          591          592          593          594          595 
##  0.067029671 -0.380118677 -0.506544503 -0.819757416 -0.766905764 
##          596          597          598          599          600 
## -0.019757416  0.452010453 -1.346183242  0.019881323 -1.019757416 
##          601          602          603          604          605 
## -0.406544503  0.119881323 -0.106544503 -0.853692851 -1.059396155 
##          606          607          608          609          610 
## -0.632970329 -1.446183242 -1.180118677  0.499158801  1.761326367 
##          611          612          613          614          615 
##  0.387752193  1.834900541  1.087752193  1.340603845  1.214178019 
##          616          617          618          619          620 
##  0.908474715  2.761326367  0.348113454  0.619881323 -0.912247807 
##          621          622          623          624          625 
##  0.993455497  0.693455497  0.967029671 -0.332970329 -0.227267025 
##          626          627          628          629          630 
##  0.819881323  0.140603845  0.846307149  0.372732975  0.646307149 
##          631          632          633          634          635 
##  0.299158801  0.519881323 -0.227267025  0.899158801  0.619881323 
##          636          637          638          639          640 
##  1.1