Handout #17Linear regressionScattergrams• A Scattergram displays the relationbetween two continuous measures (Handout#10).• It has two axes - the horizontal x-axis andthe vertical y-axis.• Each point represents two measurements ofthe same item.Liberman and Pierrehumbert (1984:174): F0 values of Peak 1 and Peak 2Prieto, van Santen and Hirschberg (1995:436): F0 peak delay (ms) by Syllableduration (ms)Relations• If two variables have a substantive relation,the datapoints in the scattergram shouldcluster around a line.• In the simplest sort of relation, that line is astraight line.• In such a case, there is a linear relationbetween the two variables.Mathematical model• The equation for a straight line in two-dimensional space is:– y = a + bx– y is the value on the y-axis– x is the value on the x-axis– b is the slope of the line, i.e. the rate of changein y as x increases one unit.– a is the y-intercept, i.e. the value for y when xis 0.Linear regression• Linear regression analysis is a method forcalculating the slope and y-intercept for the linethat comes closest to all the points.• The distance from a datapoint to the line is theerror.• Linear regression yields the line with the smallesttotal squared error for the whole dataset.• Linear regression is the appropriate method ofstatistical analysis if the dependent variable iscontinuous and some or all of the independentvariables are too.r• The correlation r is a measure of the linearrelation between two continuous variables.• n is the number of observations in the sample.• xi - x# is the deviation of a particular instance of xfrom the mean for x.• sx and sy are the standard deviations for x and y,respectively.Graphs for different values of r(Moore and McCabe 1999: 130)r• r reflects how tightly the points clusteraround the regression line.• r2 is the square of r corresponds to theproportion of the variance in y that isaccounted for by this relation with x.• Thus an r2 of .80 indicates that 80% of thevariance in y is predictible from the vcalueof x.Defining the regression line• y = a + bx• b = r * (sy / sx)• a = y# - bx#Mean height of children in Kalama,Egypt plotted against age (Moore andMcCabe 1999: 136)Regression error in the Kalama data:Moore and McCabe (1999: 140)The Kalama numbers (Moore andMcCabe 1999: 141)• x (age): x# = 23.5 mos. sx = 3.606 mos.• y (height): y# = 79.85 cm. sy = 2.302 cm.• r = 0.9944• r2 = 0.9888• b = r * (sy / sx) = 0.9944 ( 2.302/ 3.606) = 0.6348cm/mo• a = y# - bx# = 79.85 - (0.6348)(23.5) = 64.932 cm.• y = 64.932 + 0.6348xExtensions of simple linearregression (see Cohen and Cohen1983)• Multiple linear regression: More thanindependent variable• Curvilinear or nonlinear regression:Fitting to lines that other than straight ones• Logistic regression: A categoricaldependent variable and at least onecontinuous independent variableReferences• Cohen, J. and P. Cohen (1983). Applied MultipleRegression/ Correlation Analysis for the BehavioralSciences, 2nd. ed. Lawrence Erlbaum Associates,Hillsdale, N.J.• Liberman, M. and J. Pierrehumbert (1984). IntonationalInvariance Under Changes in Pitch Range and Length. InM. Aronoff and R. Oehrle (eds.) Language SoundStructure. MIT Press, Cambridge. 157-233.• Moore, D. and G. McCabe (1999). Introduction to thePractice of Statistics. W.H.Freeman and Co., New York.• Prieto, P., J. van Santen and J. Hirschberg (1995). TonalAlignment Patterns in Spanish. Journal of Phonetics