bivariate regression

Bivariate regression

the correlation coefficient measures the association between 2 sets of paired variates, but it does not

1) tell us the way the two variables are related

2) does not allow us to predict the value of one variable with knowledge of the value of the other variable

3) doesn’t signal anomalies in the relationship between individual pairs

bivariate regression lets us do all of these things

dependent and independent variables

regression allows us to suggest (hypothesize) causal relationships and their direction - substantiated by previous research and common sense

scattergram - used to plot dependent along y axis, independent along x axis

regression involves plotting a ‘best-fit’ line between the points on a scattergram

convention is to treat the dependent variable as PREDICTED and the independent variable as the PREDICTOR

prediction/interpolation is one of the main uses because x and y are sampled. As we don’t have complete information on values for a given x we want to interpolate intermediate values from the best fit line on the scattergram

derivation of best fit line

example 1:

y	2	6	8	14	22
x	1	3	4	7	11

easy to place ‘best’ line through these points as the association is perfect

correlation coefficient =1

there are no residuals/anomalies/no deviations of points from general relationship since every point is on the regression line

however variables are rarely perfectly correlated because of 1) poor/theory/understanding or 2) measurement error

example 2

can place ‘best-fit’ line through points although r<1 and so points representing variates do form a straight line

deviations/anomalies/residuals from regression are shown as 89

residuals: why plot them vertically rather than perpendicular to the regression line?

Because residuals are the difference between the actual/observed values of the dependent variable (y values) and the expected/predicted value of the dependent variable (y hat ) for a particular value of x

fitting the regression line by least square method

any straight line drawn on an x y coordinate system can be represented by an equation of the form

least square- objective to find the combination of a,b values which minimize the sum of squares of the residual values, that is, minimize the difference between the actual and predicted values at particular values of x

example

Gy_i=247	Gx_i=14,378	G(yi-y{_I)²=4095.8
	(Gx_i)²=206,726,884	Gx_i²=50,744,856
	Gx_iy_i=1,141,072.8

river	discharge (y)	catchment (x)	y hat=a+b prediction	(y-y hat) residual
Yellow	3.3	672	-6.0	9.3
Ganges	11.7	956	2.5	9.2
Amazon	175	5775	146.5	28.5
Mississippi	18.4	3269	71.6	-53.2
Mekong	11	795	-2.3	13.3
Indus	5.6	969	2.9	2.7
Yangtze	22	1942	31.9	-9.9

y hat=a+bx	I=1	-26.09+[0.02988*672]	=	-6.01
	I=2	-26.09+[0.02988*956]	=	2.48
	I=3	-26.09+[0.02988*5775]	=	146.48

the standard method of measuring the goodness of fit of a regression is to calculate the extent to which the regression accounts for the variation in the observed values of the dependent variable

this is done by calculating the variance of the observed value of y

tests on the residuals

a complementary test of goodness of fit involves looking at the residuals

there should be no systematic variation in the residuals

example of coefficient of determination

river	discharge (y)	y²	catchment (x)	y hat=a+b prediction	y hat²	(y-y hat) residual
Yellow	3.3	10.89	672	-6.0	36	9.3
Ganges	11.7	136.89	956	2.5	6.25	9.2
Amazon	175	30625	5775	146.5	21462.25	28.5
Mississippi	18.4	338.56	3269	71.6	5126.56	-53.2
Mekong	11	121	795	-2.3	5.29	13.3
Indus	5.6	31.36	969	2.9	8.41	2.7
Yangtze	22	484	1942	31.9	1017.6	-9.9
Total	247	31747.7		247.1	27662.36