We often want to know whether various variables are ‘linked’, i.e., correlated. This can be interesting in itself, but is also important if we want to predict one variable’s value given a value of the other.
means that correlation cannot be validly used to infer a causal relationship between the variables not be taken to mean that correlations cannot indicate causal relations. However, the causes underlying the correlation, if any, may be indirect and unknown A correlation between age and height in children is fairly causally transparent, but a correlation between mood and health in people is less so. Does improved mood lead to improved health? Or does good health lead to good mood? Or does some other factor underlie both? Or is it pure coincidence? In other words, a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the causal relationship, if any, might be.
Variance is just a definition. The reason we’re squaring it is to have it get a positive value whether dx is negative or positive, so that we can sum them and positives and negatives will not cancel out. Variance is spread around a mean, covariance is the measure of how much x and y change together; very similar: multiply 2 variables rather than square 1
Problem with Covariance: The value obtained by covariance is dependent on the size of the data’s standard deviations: if large, the value will be greater than if small… even if the relationship between x and y is exactly the same in the large versus small standard deviation datasets.
Can only compare covariances between different variables to see which is greater.
The distance of r from 0 indicates strength of correlation r = 1 or r = (-1) means that we can predict y from x and vice versa with certainty; all data points are on a straight line. i.e., y = ax + b The correlation is 1 in the case of an increasing linear relationship, −1 in the case of a decreasing linear relationship, and some value in between in all other cases, indicating the degree of linear dependence between the variables. The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables. the correlation coefficient detects only linear dependencies between two variables
X av = 3, y av = 2 If extreme value such as y = 5 is possible (graph) X – 1,2,3,4,5, Y – 1,2,3,4,0
the interpretation of a correlation coefficient depends on the context and purposes. A correlation of 0.9 may be very low if one is verifying a physical law using high-quality instruments, but may be regarded as very high in the social sciences where there may be a greater contribution from complicating factors.
So to understand the relationship between two variables, we want to draw the ‘best’ line through the cloud – find the best fit . This is done using the principle of least squares
T tests anovas and regression
T tests, ANOVAs and regression Tom Jenkins Ellen MeierottoSPM Methods for Dummies 2007
T tests in SPM: Did the observed signalchange occur by chance or is it stat.significant? Recall GLM. Y= X β + ε β1 is an estimate of signal change over time attributable to the condition of interest Set up contrast (cT) 1 0 for β1: 1xβ1+0xβ2+0xβn/s.d Null hypothesis: cTβ=0 No significant effect at each voxel for condition β1 Contrast 1 -1 : Is the difference between 2 conditions significantly non-zero? t = cTβ/sd[cTβ] – 1 sided
ANOVA Variances not means Total variance= model variance + error variance Results in F score- corresponding to a p valueVariance n ∑ ( xi − x ) 2 F test = Model variance /Error variances2 = i =1 n −1
Partitioning the varianceGroup Group Group Group Group Group1 2 1 2 1 2 Total = Model + Error (Between groups) (Within groups)
T vs F tests F tests- any differences between multiple groups, interactions Have to determine where differences are post-hoc SPM- T- one tailed (con) SPM- F- two tailed (ess)
Conclusions T tests describe how unlikely it is that experimental differences are due to chance Higher the t score, smaller the p value, more unlikely to be due to chance Can compare sample with population or 2 samples, paired or unpaired ANOVA/F tests are similar but use variances instead of means and can be applied to more than 2 groups and other more complex scenarios
Acknowledgements MfD slides 2004-2006 Van Belle, Biostatistics Human Brain Function Wikipedia
Topics Covered: Is there a relationship between x and y? What is the strength of this relationship Pearson’s r Can we describe this relationship and use it to predict y from x? Regression Is the relationship we have described statistically significant? F- and t-tests Relevance to SPM GLM
Relationship between x and y Correlation describes the strength and direction of a linear relationship between two variables Regression tells you how well a certain independent variable predicts a dependent variable CORRELATION ≠ CAUSATION In order to infer causality: manipulate independent variable and observe effect on dependent variable
Scattergrams Y Y Y Y Y Y X X XPositive correlation Negative correlation No correlation
Variance vs. Covariance Do two variables change together? nVariance ~ ∑(x i − x) 2 S = 2 x i =1 DX * DX n nCovariance ~ ∑(x i − x)( yi − y ) DX * DY cov( x, y ) = i =1 n
Covariance n ∑(x i − x)( yi − y ) cov( x, y ) = i =1 n When X and Y : cov (x,y) = pos. When X and Y : cov (x,y) = neg. When no constant relationship: cov (x,y) =0
Example Covariance76 x y xi − x yi − y ( xi − x )( yi − y )5 0 3 -3 0 04 2 2 -1 -1 132 3 4 0 1 01 4 0 1 -3 -30 6 6 3 3 9 0 1 2 3 4 5 6 7 x=3 y=3 ∑= 7 n ∑(x i − x)( yi − y )) 7 What does thiscov( x, y ) = i =1 = = 1.4 number tell us? n 5
Example of how covariance value relies on variance High variance data Low variance data Subject x y x error * y x y X error * y error error 1 101 100 2500 54 53 9 2 81 80 900 53 52 43 61 60 100 52 51 14 51 50 0 51 50 05 41 40 100 50 49 16 21 20 900 49 48 47 1 0 2500 48 47 9Mean 51 50 51 50Sum of x error * y error : 7000 Sum of x error * y error : 28Covariance: 1166.67 Covariance: 4.67
Pearson’s R − ∞ ≤ cov( x, y ) ≤ ∞ Covariance does not really tell us anything Solution: standardise this measure Pearson’s R: standardise by adding std to equation: cov( x, y ) rxy = sx s y
Basic assumptions Normal distributions Variances are constant and not zero Independent sampling – no autocorrelations No errors in the values of the independent variable All causation in the model is one-way (not necessary mathematically, but essential for prediction)
Pearson’s R: degree of linear dependence n n ∑ (x i − x)( yi − y ) ∑ ( x − x)( y − y) i icov( x, y ) = i =1 rxy = i =1 n nsx s y −1 ≤ r ≤ 1 n ∑Z xi * Z yi rxy = i =1 n
Limitations of r ˆ r is actually r r = true r of whole population ˆ r = estimate of r based on data r is very sensitive to extreme values: 5 4 3 2 1 0 0 1 2 3 4 5 6
In the real world… r is never 1 or –1 interpretations for correlations in psychological research (Cohen)Correlation Negative PositiveSmall -0.29 to -0.10 00.10 to 0.29Medium -0.49 to -0.30 0.30 to 0.49Large -1.00 to -0.50 0.50 to 1.00
Regression Correlation tells you if there is an association between x and y but it doesn’t describe the relationship or allow you to predict one variable from the other. To do this we need REGRESSION!
Best-fit Line Aim of linear regression is to fit a straight line, ŷ = ax + b, to data that gives best prediction of y for any value of x This will be the line that ŷ = ax + b minimises distance between slope intercept data and fitted line, i.e. the residuals ε = ŷ, predicted value = y i , true value ε = residual error
Least Squares Regression To find the best line we must minimise the sum of the squares of the residuals (the vertical distances from the data points to our line) Model line: ŷ = ax + b a = slope, b = intercept Residual (ε) = y - ŷ Sum of squares of residuals = Σ (y – ŷ)2 we must find values of a and b that minimise Σ (y – ŷ)2
Finding b First we find the value of b that gives the min sum of squares b ε b ε b Trying different values of b is equivalent to shifting the line up and down the scatter plot
Finding a Now we find the value of a that gives the min sum of squares b b b Trying out different values of a is equivalent to changing the slope of the line, while b stays constant
Minimising sums of squares Need to minimise Σ(y–ŷ)2 ŷ = ax + b so need to minimise: sums of squares (S) Σ(y - ax - b)2 If we plot the sums of squares for all different values of a and b we get a parabola, because it is a squared term Gradient = 0 min S Values of a and b So the min sum of squares is at the bottom of the curve, where the gradient is zero.
The maths bit So we can find a and b that give min sum of squares by taking partial derivatives of Σ(y - ax - b)2 with respect to a and b separately Then we solve these for 0 to give us the values of a and b that give the min sum of squares
The solution Doing this gives the following equations for a and b: r sy r = correlation coefficient of x and y a= s sy = standard deviation of y x sx = standard deviation of x You can see that: A low correlation coefficient gives a flatter slope (small value of a) Large spread of y, i.e. high standard deviation, results in a steeper slope (high value of a) Large spread of x, i.e. high standard deviation, results in a flatter slope (high value of a)
The solution cont. Our model equation is ŷ = ax + b This line must pass through the mean so: y = ax + b b = y – ax We can put our equation into this giving: r sy r = correlation coefficient of x and y b=y- x sy = standard deviation of y sx sx = standard deviation of x The smaller the correlation, the closer the intercept is to the mean of y
Back to the model We can calculate the regression line for any data, but the important question is: How well does this line fit the data, or how good is it at predicting y from x?
How good is our model? ∑(y – y)2 SSy Total variance of y: sy2 = = n-1 dfy Variance of predicted y values (ŷ): ∑(ŷ – y)2 SSpred This is the variance sŷ2 = = explained by our n-1 dfŷ regression model Error variance: This is the variance of the error between our predicted y values ∑(y – ŷ)2 SSer and the actual y values, and serror = 2 = thus is the variance in y that is n-2 dfer NOT explained by the regression model
How good is our model cont. Total variance = predicted variance + error variance sy2 = sŷ2 + ser2 Conveniently, via some complicated rearranging sŷ2 = r2 sy2 r2 = sŷ2 / sy2 so r2 is the proportion of the variance in y that is explained by our regression model
How good is our model cont. Insert r2 sy2 into sy2 = sŷ2 + ser2 and rearrange to get: ser2 = sy2 – r2sy2 = sy2 (1 – r2) From this we can see that the greater the correlation the smaller the error variance, so the better our prediction
Is the model significant? i.e. do we get a significantly better prediction of y from our regression equation than by just predicting the mean? F-statistic: complicated rearranging sŷ2 r2 (n - 2)2 F(df ,df ) = =......= ŷ er ser2 1 – r2 And it follows that: r (n - 2) So all we need to(because F = t 2) t(n-2) = know are r and n ! √1 – r2
General Linear Model Linear regression is actually a form of the General Linear Model where the parameters are a, the slope of the line, and b, the intercept. y = ax + b +ε A General Linear Model is just any model that describes the data in terms of a straight line
Multiple regression Multiple regression is used to determine the effect of a number of independent variables, x1, x2, x3 etc., on a single dependent variable, y The different x variables are combined in a linear way and each has its own regression coefficient: y = a1x1+ a2x2 +…..+ anxn + b + ε The a parameters reflect the independent contribution of each independent variable, x, to the value of the dependent variable, y. i.e. the amount of variance in y that is accounted for by each x variable after all the other x variables have been accounted for
SPM Linear regression is a GLM that models the effect of one independent variable, x, on ONE dependent variable, y Multiple Regression models the effect of several independent variables, x1, x2 etc, on ONE dependent variable, y Both are types of General Linear Model GLM can also allow you to analyse the effects of several independent x variables on several dependent variables, y1, y2, y3 etc, in a linear combination This is what SPM does and will be explained soon…