More on regression Species richness pH Humidity Light Temper ature Organic matter content Species richness pH Humidity Light Temper ature Organic matter.

More on regression

Species richness

pH

Humidity

Light

Temperature

Organic matter content

Species richness

pH

Humidity

Light

Temperature

Organic matter content

Is it possible to infer causal relationships between

model drivers from regression analysis?

Is it possible to compare the goodness of different

models?

Is it possible to quantify the influence of different drivers?

Path analysis and linear structure models (Structure equation modelling SEM)

Y

X3X2 X4X1

e

Multiple regression

YX3

X2

X4X1

ee

e

e

e

Path analysis tries to do something that is logically impossible, to derive causal relationships from sets of observations.

Path analysis defines a whole model and tries to separate correlations into direct and indirect effects

eXaXaXaXaaY 443322110

The error term e contains the part of the variance in Y that is not explained by the model. These errors are called residuals

Regression analysis does not study the relationships between the predictor

variables

X Z

Y

WpXW pZX

pZYpXY

e e

e

e

Path analysis is largely based on the computation of partial coefficients of correlation.

Path coefficients

Path analysis is a model confirmatory tool. It should not be used to generate models or even to seek for models that fit the data set.

xw

xy

zx zy

W p X e

X p Y e

Z p X p Y e

xw

xy

zx zy

p X W e 0

X p Y e 0

p X p Y Z e 0

We start from regression functions

Using Z-transformed values we get

X

W

Z

Y

pXW

pYX

pXZ pYZ

W xw X

X xy Y

Z zx X zy Y

W Y xw X Y Y

X W xy Y W W

Z W zx X W zy Y W W

X Z xy Y X X

X Y xy Y Y Y

Z Y zx X Y zy Y Y Y

WY xw XY

XW xy YW

ZW zx XW zy YW

XZ xy YX

XY

Z p Z e

Z p Z e

Z p Z p Z e

Z Z p Z Z eZ

Z Z p Z Z eZ

Z Z p Z Z p Z Z eZ

Z Z p Z Z eZ

Z Z p Z Z eZ

Z Z p Z Z p Z Z eZ

r p r

r p r

r p r p r

r p r

r

xy

ZY zx XY zy

p

r p r p

eZY = 0

ZYZY = 1

ZXZY = rXY

xw

xy

zx zy

p X W e 0

X p Y e 0

p X p Y Z e 0

Path analysis is a nice tool to generate hypotheses.It fails at low coefficients

of correlation and circular model structures.

𝒀=𝑹𝑷

Species richness and soil characteristics of ground beetles

Species richness

pH

Humidity

Light

Temperature

Organic matter content

pLH

pLT

pTH

pPHO

pTSpHS pOS

𝑆=𝑝𝐻𝑆𝐻+𝑝𝑂𝑆𝑂+𝑝𝑇𝑆𝑇

𝐻=𝑝𝐿𝐻 𝐿+𝑝𝑇𝐻𝑇

𝑇=𝑝𝐿𝑇 𝐿

𝑂=𝑝𝑃𝐻𝑂 𝑃𝐻

𝑆𝐻=𝑝𝐻𝑆𝐻𝐻+𝑝𝑂𝑆𝑂𝐻+𝑝𝑇𝑆𝑇𝐻𝑆𝑇=𝑝𝐻𝑆𝐻𝑇+𝑝𝑂𝑆𝑂𝑇+𝑝𝑇𝑆𝑇𝑇𝑆𝑂=𝑝𝐻𝑆𝐻𝑂+𝑝𝑂𝑆𝑂𝑂+𝑝𝑇𝑆𝑇𝑂

𝐻𝑇=𝑝𝐿𝐻 𝐿𝑇+𝑝𝑇𝐻𝑇𝑇𝐻𝑂=𝑝𝐿𝐻 𝐿𝑂+𝑝𝑇𝐻 𝑇𝑂

𝑇𝑂=𝑝𝐿𝑇 𝐿𝑂

𝑆𝑆=𝑝𝐻𝑆𝐻𝑆+𝑝𝑂𝑆𝑂𝑆+𝑝𝑇𝑆𝑇𝑆

TWe formulate a model of causal relationships.We multiply each equation by the other variables.

WE have seven unknowns and need seven linear equations.

𝑆𝐻=𝑝𝐻𝑆𝐻𝐻+𝑝𝑂𝑆𝑂𝐻+𝑝𝑇𝑆𝑇𝐻𝑆𝑇=𝑝𝐻𝑆𝐻𝑇+𝑝𝑂𝑆𝑂𝑇+𝑝𝑇𝑆𝑇𝑇𝑆𝑂=𝑝𝐻𝑆𝐻𝑂+𝑝𝑂𝑆𝑂𝑂+𝑝𝑇𝑆𝑇𝑂

𝐻𝑇=𝑝𝐿𝐻 𝐿𝑇+𝑝𝑇𝐻𝑇𝑇𝐻𝑂=𝑝𝐿𝐻 𝐿𝑂+𝑝𝑇𝐻 𝑇𝑂

𝑇𝑂=𝑝𝐿𝑇 𝐿𝑂T

Species Light Temprature pHOrganic matter content

Humidity

10 3.97 3.42 4.16 4.30 3.3518 3.17 3.37 4.13 3.70 3.2125 3.38 3.40 4.05 3.75 3.3825 3.01 3.45 4.09 3.91 3.1829 2.67 3.30 3.93 3.95 3.0940 2.95 3.42 4.03 3.90 3.1720 3.04 3.30 4.06 4.01 3.0819 2.86 3.54 4.23 3.93 3.0523 3.24 3.31 4.14 4.12 3.0730 2.96 3.40 4.08 3.94 3.0523 3.74 3.52 4.25 4.14 3.0731 3.02 3.30 3.83 3.82 3.2714 3.09 3.42 3.96 3.89 3.1316 3.81 3.45 4.34 4.19 3.1424 3.68 3.49 4.33 3.38 3.4116 3.67 3.48 4.03 3.92 3.3713 3.47 3.44 3.93 3.55 3.34

Species Light Temprature pH Organic_matter_content Humidity

Species 0 0.019219 0.27591 0.34139 0.58463 0.36388Light -0.56068 0 0.07543 0.033355 0.56839 0.022344

Temprature -0.28026 0.44232 0 0.011559 0.71838 0.52892pH -0.24591 0.51755 0.59609 0 0.49036 0.75606

Organic_matter_content -0.14277 0.14892 -0.094463 0.1796 0 0.040297

Humidity -0.23502 0.54942 0.16418 -0.081422 -0.50144 0

Correlation matrix

Y pHS pOS pTS pLH pTH pLT pPHOrSH 1 rOH rTH 0 0 0 0rST rHT rOT 1 0 0 0 0rSO rHO 1 rTO 0 0 0 0rHT 0 0 0 rLT 1 0 0rHO 0 0 0 rLO rTO 0 0rTO 0 0 0 0 0 rLO 0rOT 0 0 0 0 0 0 rPHT

Y pHS pOS pTS pLH pTH pLT pPHO-0.23502 1 0.040297 0.16418 0 0 0 0-0.28026 0.52892 0.71838 1 0 0 0 0-0.14277 0.040297 1 -0.09446 0 0 0 00.52892 0 0 0 0.44232 1 0 0

0.040297 0 0 0 0.14892 -0.09446 0 0-0.09446 0 0 0 0 0 0.14892 00.71838 0 0 0 0 0 0 0.011559

𝑆𝐻=𝑝𝐻𝑆𝐻𝐻+𝑝𝑂𝑆𝑂𝐻+𝑝𝑇𝑆𝑇𝐻𝑆𝑇=𝑝𝐻𝑆𝐻𝑇+𝑝𝑂𝑆𝑂𝑇+𝑝𝑇𝑆𝑇𝑇𝑆𝑂=𝑝𝐻𝑆𝐻𝑂+𝑝𝑂𝑆𝑂𝑂+𝑝𝑇𝑆𝑇𝑂

𝐻𝑇=𝑝𝐿𝐻 𝐿𝑇+𝑝𝑇𝐻𝑇𝑇𝐻𝑂=𝑝𝐿𝐻 𝐿𝑂+𝑝𝑇𝐻 𝑇𝑂

𝑇𝑂=𝑝𝐿𝑇 𝐿𝑂T

pHS -0.21888pOS -0.13999pTS -0.06392pLH 0.473304pTH 0.319568pLT -0.63432pPHO 62.14897

𝑷=𝑹−𝟏𝒀

Species richness

pH

Humidity

Light

Temperature

Organic matter content

0.47

-0.63

0.31

62.2

-0.06-0.22 -0.14

𝒀=𝑹𝑷

N X A B C1 1.00 0.68 0.55 2.162 1.30 0.98 1.49 0.453 1.42 0.74 0.13 0.554 1.70 0.12 0.28 2.345 2.47 0.63 0.73 0.606 3.02 0.73 1.73 0.147 3.91 0.19 0.28 2.608 4.42 0.73 1.36 2.749 5.09 1.91 1.89 0.9910 5.27 1.49 0.96 1.2111 5.58 1.11 1.14 3.2012 6.34 0.84 1.31 1.0113 6.64 1.72 2.57 0.9214 7.32 0.87 1.17 3.21

R2 is the explained variance in abivariate comparison

Logistic and other regression techniques

01

n

i ii

Y a a x

We use odds

The logistic regression model

𝑍= 𝑒𝑌

1+𝑒𝑌

𝑌=𝑙𝑛( 𝑝1−𝑝 )=𝑎0+∑𝑎𝑖 𝑥𝑖

𝑝1−𝑝

=𝑒𝑎0+∑ 𝑎𝑖𝑥 𝑖  

𝑝= 𝑒𝑎0+∑ 𝑎𝑖𝑥 𝑖  

1+𝑒𝑎0+∑ 𝑎𝑖 𝑥 𝑖  

P defines a probability according to a logistic model

1

0.5

Threshold

Surely malesSurely females

P

Gender A B CFemale 0.038 0.165 2.211Female 0.500 0.987 2.894Female 0.864 0.759 0.860Female 0.590 1.071 2.434Female 0.385 0.749 0.984Female 0.703 0.879 2.745Female 0.629 1.047 2.774Male 0.730 0.798 2.951Male 1.367 1.841 3.174Male 1.325 0.850 1.337Male 0.958 1.551 3.000Male 1.173 1.164 1.077Male 1.559 1.521 3.266Male 1.027 1.251 3.315X 0.900 0.856 2.345

0

0.2

0.4

0.6

0.8

1

Fem

ale

Fem

ale

Fem

ale

Fem

ale

Fem

ale

Fem

ale

Fem

ale

Mal

eM

ale

Mal

eM

ale

Mal

eM

ale

Mal

e

P

Gender

Gender A B CX 0.900 0.856 2.345a0 a1 a2 a3 a0

37.425 2.900 8.000 -52.5Y 2.436eY 11.43p 0.92 X is with probability 0.92 a male.

𝑝= 𝑒𝑎0+∑ 𝑎𝑖𝑥 𝑖  

1+𝑒𝑎0+∑ 𝑎𝑖 𝑥 𝑖  

Regression trees

Region AMT TAR RAI RAR

Annual mean

temperature

Temperature range

Annual mean precipitation

Precipitation range

Argentina_South 7.3 27.3 217 30

Argentina_South 7.9 25.7 375 66Argentina_South 7.2 24.4 568 94Argentina_South 7.1 23.8 685 104Argentina_South 7.4 26.5 284 48Argentina_South 7.8 25.3 416 74Argentina_Pampas 15 30.2 363 33Argentina_Pampas 15.1 31 342 32Argentina_Pampas 15.2 31.6 320 30Argentina_Pampas 15.2 32.2 313 26Argentina_Pampas 14.7 32.7 275 27Argentina_Pampas 14.4 32.5 194 17Argentina_East 18.6 31.8 243 51Argentina_East 19.2 30 355 73

Root

Australia Central Other

12 29RAR < 14.5

AMT < 11.15

Argentina South

Other

6 23

AMT < 16.45

Argentina Pampas

Other

6 17

RAI < 380

Argentina East

Other

6 11

Regression tree analysis tries to groups cases according to predefined nominal and ordinal variables and returns variables levels that best group these cases.It uses a heuristic pattern seeking algorithm.

N X A B1 1.00 0.68 0.552 1.30 0.98 1.493 1.42 0.74 0.134 1.70 0.12 0.285 2.47 0.63 0.736 3.02 0.73 1.737 3.91 0.19 0.288 4.42 0.73 1.369 5.09 1.91 1.8910 5.27 1.49 0.9611 5.58 1.11 1.1412 6.34 0.84 1.3113 6.64 1.72 2.5714 7.32 0.87 1.17

What is the correlation between B and X?

y = 0.17x + 0.43R² = 0.29

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8

XB

What is the pure correlation between B and X excluding the influence of A on both X and B?

We need the partial correlation of X and B.

A B

X

rAB

rBXrAX

y = 0.12x + 0.43

0

0.5

1

1.5

2

0 2 4 6 8

X

A

y = 0.99x + 0.21

0

0.5

1

1.5

2

2.5

3

0 1 2 3

B

A

DB

DX

y = 0.05x - 0.18R² = 0.06

-1

-0.5

0

0.5

1

0 2 4 6 8

DB =

X

DX = B

Partial regressions are the regression of residuals excluding a third factor.

/ 2 21 1

XY XZ YZ

XY Z

XZ YZ

r r rr

r r

r\p X A B CX 0 0.29371 0.17742 0.03325A 0.33073 0 0.12024 0.3568B 0.41704 0.4732 0 0.27957C 0.61517 -0.29216 -0.33999 0

Coeff. Std.err. r2 pConstant -0.561 1.317 0.679A 1.425 1.286 0.109 0.294B 1.388 0.957 0.174 0.177C 1.065 0.432 0.378 0.033

R^20.0000.2620.2910.094

Partial linear correlations

The partial linear correlations of A, B, and C on X.

To show the isolated influence of single predictors we show the squared partial correlation coefficients within linear

regression results.

Multiple regression results