0

0.5

1

1.5

2

2.5

-1 1 3 5

Mean grain size, phi

Sort

ing

Braided stream

Point bar

Stream mouth bar

Tidal flat

Inter distrib. beach

Mature beach

Dune

GY2100 Geographical Data Analysis

Lecture 4Regression analysis and

statistical inference

DEPARTMENT OF GEOGRAPHY

The statistical utility of a regression line

The regression model and its underlying assumptions

Yi i = + Xi i + ii

Systematic or deterministic component represented by a straight line

Random or stochastic component represented by the deviations of the observations about the line

– alpha

– beta

delta

Illustrating regression using the

fixed X model

Assumptions of the regression model

1. The relationship between X and Y is linear;

2. Values of X are fixed and measured without error;

3. The disturbance terms i are normally distributed with equal variance about the line Y = + X and each has an expected value E(i) = 0. This means that the expected value for a given value of X is E(Yi,Xi) = + Xi;

Assumptions of the regression model

4. The di are statistically uncorrelated:

a. There is no autocorrelation (di term is uncorrelated with X)

b. There is no spatial autocorrelation (di are not correlated with another variable).

Testing the assumptions

1. Specific statistical tests using residuals of the sample regression line as estimates of the error term in the true population regression model

2. Histogram of residuals3. Examination of residual plots

Examination of residual plots

Recap

Yi = a + bXi + di

Inferences in regression analysis

Slope parameter () Intercept parameter () Precision of estimates derived from

the sample regression equation

Testing if =0

Elevation above sea-level vs mean annual rainfallfor Scotland

Y' = 2.38X + 895

0

500

1000

1500

2000

2500

0 100 200 300 400 500 600

Elevation (m)

Rai

nfal

l (m

m y

r-1)

If = 0 then

Y’= constant for all X

Testing =0 using the t test

HypothesesHo: = 0

H1: 0

Under Ho, repeated sampling yields a distribution of b which follows a t distribution about an expected value of = 0.

Testing =0 using the t test

Since we are testing

= 0, then

bsb

nt

2

n

i

n

iii

yx

nXX

sbs

1

2

1

2 1

where

df = n-2 is the number of

degrees of freedom

sb = estimated standard error

of the sampling distribution of b

bn s

bt

2

The test statistic to be

calculated is

Regression in EXCELRegression Statistics

Multiple R 0.78

R square 0.61

Adj R square 0.59

Standard error 242.79

Observations 20

ANOVA

df SS MS F Sig F

Regression 1 1691558 1691558

28.7 4.31E-5

Residual 18 1061062 58948

Total 19 2752620

Coeffs Standard Error t stat P-value Lower 95% Upper 95%

Intercept 895 149.76 5.98 1.178E-5

581 1210

Elevation, m 2.38 0.444 5.36 4.31E-5 1.44 3.31

Testing =0 using the t test

Since we are testing

= 0, then

bsb

nt

2

bn s

bt

2

The test statistic to be

calculated is

So……

36.5444.038.2 t

With df=n-2=18,

tcrit = 2.1 at = 0.05.

Regression in EXCELRegression Statistics

Multiple R 0.78

R square 0.61

Adj R square 0.59

Standard error 242.79

Observations 20

ANOVA

df SS MS F Sig F

Regression 1 1691558 1691558

28.7 4.31E-5

Residual 18 1061062 58948

Total 19 2752620

Coeffs Standard Error t stat P-value Lower 95% Upper 95%

Intercept 895 149.76 5.98 1.178E-5 581 1210

Elevation, m 2.38 0.444 5.36 4.31E-5 1.44 3.31

Testing =Q using the t test(Q0)

bn s

bt

2

The test statistic to be

calculated is

If we are testing = 4, then

444.0438.2

2

bn s

bt

Testing =0 using the F test

Decomposition of the variance using sums of squares

Total sum = Regression sum + Residual sum

of squares of squares of squares

n

iii

n

i

n

iii YYYYYYTSS

1

2

1 1

22)'()'()(

Testing =0 using the F test

Decomposition of the variance using sums of squaresTotal sum of squares (TSS) is the sum of the squared deviations of the individual observations about their mean

n

ii YYTSS

1

2)(

Regression sum of squares (RSS) is the sum of the squared deviations of the predicted Y values (Y’) about the mean Y

n

ii YYRSS

1

2)'(

The difference between TSS and RSS is ‘unexplained’ by the regression line and

is therefore the residual sum of squares (Residual SS)

n

iii YYSSsid

1

2)'(Re

Testing =0 using the F test

Decomposition of

the variance using

sums of squares

TSS=RSS+Resid SS

n

ii YYTSS

1

2)(

n

ii YYRSS

1

2)'(

n

iii YYSSsidual

1

2)'(Re

Regression in EXCELRegression Statistics

Multiple R 0.78

R square 0.61

Adj R square 0.59

Standard error 242.79

Observations 20

ANOVA

df SS MS F Sig F

Regression 1 1691558 1691558

28.7 4.31E-5

Residual 18 1061062 58948

Total 19 2752620

Coeffs Standard Error t stat P-value Lower 95% Upper 95%

Intercept 895 149.76 5.98 1.178E-5

581 1210

Elevation, m 2.38 0.444 5.36 4.31E-5 1.44 3.31

Regression in Excel

ANOVA

df SS MS F Sig F

Regression

1 RSS RSS/df RMSS/Resid MSS

Residual n-2 Resid SS Resid SS/df

Total n-1 TSS TSS/df

RMSS = Regression MSS

Resid MSS – Residual MSS

Regression in Excel

ANOVA

df SS MS F Sig F

Regression 1 1691558 1691558 28.7 4.31E-5

Residual 18 1061062 58948

Total 19 2752620

With 1 and 18 df,

Fcrit 2.7 at = 0.05.

Constructing a confidence interval for

In general bb stbstb ..

For the rainfall data

31.344.1

444.0*1.238.2444.0*1.238.2

Regression in EXCELRegression Statistics

Multiple R 0.78

R square 0.61

Adj R square 0.59

Standard error 242.79

Observations 20

ANOVA

df SS MS F Sig F

Regression 1 1691558 1691558

28.7 4.31E-5

Residual 18 1061062 58948

Total 19 2752620

Coeffs Standard Error t stat P-value Lower 95% Upper 95%

Intercept 895 149.76 5.98 1.178E-5

581 1210

Elevation, m 2.38 0.444 5.36 4.31E-5 1.44 3.31