Some Analysis of Some Perch Catch Data

56 perch were caught in a freshwater lake in Finland

Their weights, lengths, heights and widths were recorded

It may be anticipated that thefish's weights depend on their lengths, heights and widths whose product is a proxy for volume

Some questions/goals:

summary

outliers

prediction

interpretation of coefficients

linear

gaussian errors

preparation for a comparative study

presentation of results

...

Some of the data.

Weight(g) Length(cm) Height(cm) Width(cm)

5.9 8.4 2.11 1.41

32.0 13.7 3.53 2.00

40.0 15.0 3.82 2.43

51.5 16.2 4.59 2.63

70.0 17.4 4.59 2.94

100.0 18.0 5.22 3.32

78.0 18.7 5.20 3.12

summary(weight)

Min. 1st Qu. Median Mean 3rd Qu. Max.

5.9 120.0 207.5 382.2 692.5 1100.0

stem() The decimal point is 2 digit(s) to the right of the |

0 | 134578899011222333345555789

2 | 0235567002

4 | 16

6 | 59900

8 | 224500

10 | 000200

The decimal point is 2 digit(s) to the right of the |

0 | 134578899

1 | 011222333345555789

2 | 0235567

3 | 002

4 |

5 | 16

6 | 599

7 | 00

8 | 2245

9 | 00

10 | 0002

11 | 00

ecdf()

qqnorm()

density()

boxplot()

boxplot()

library(lattice) splom

plot()

qqnorm()

summary(junk2)

Call:

lm(formula = logweight ~ loglength + logheight + logwidth)

Residuals:

Min 1Q Median 3Q Max

-0.075575 -0.022514 -0.001842 0.022046 0.091880

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -1.0060 0.1690 -5.953 2.28e-07 ***

loglength 1.6197 0.2265 7.151 2.84e-09 ***

logheight 0.8226 0.2167 3.796 0.000386 ***

logwidth 0.5622 0.1803 3.119 0.002958 **

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.03767 on 52 degrees of freedom

Multiple R-squared: 0.994, Adjusted R-squared: 0.9937

F-statistic: 2890 on 3 and 52 DF, p-value: < 2.2e-16

qqnorm()

anova(junk2)

Analysis of Variance Table

Response: logweight

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

loglength 1 12.2353 12.2353 8623.0612 < 2.2e-16 ***

logheight 1 0.0534 0.0534 37.6351 1.179e-07 ***

logwidth 1 0.0138 0.0138 9.7278 0.002958 **

Residuals 52 0.0738 0.0014

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

logw = -1.0060 + 1.6197 logl + .8226 logh + .5622 logw

(.1690) (.2265) (.2167) (.1803)

hi library(MASS) lm.influence()$hat

Ei* = Ei /[S(-i) (1-hi )] qqline(studres())

Ei* [hi /(1-hi )] dffits

Di cooks.distance

library(car) av.plots()

junk3<-cbind(length-mean(length),width-mean(width),height-mean(height))

cor(junk3)

[,1] [,2] [,3]

[1,] 1.0000000 0.9746171 0.9855836

[2,] 0.9746171 1.0000000 0.9829435

[3,] 0.9855836 0.9829435 1.0000000

Is X'X near singular?

Would make interpretation of coefficients difficult

junk3<-cbind(length-mean(length),width-mean(width),height-mean(height))

junk4<-svd(junk3)

junk4$d

junk4$d

[1] 71.313882 3.927869 2.050682

Conclusions.

Can replace weight by product of lengths

Traditional linear model results not strongly invalidated

Began with EDA, to look for unusual "things", then moved onto linear model

...