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t-tests, ANOVA and regression - and their application to the statistical analysis of fMRI data Lorelei Howard and Nick Wright MfD 2008 Overview • • • • Why do we need statistics? P values T-tests ANOVA Why do we need statistics? • To enable us to test experimental hypotheses – H0 = null hypothesis – H1 = experimental hypothesis • In terms of fMRI – Null = no difference in brain activation between these 2 conditions – Exp = there is a difference in brain activation between these 2 conditions 2 types of statistics • Descriptive Stats – e.g., mean and standard deviation (S.D) • Inferential statistics – t-tests, ANOVAs and regression Issues when making inferences • So how do we know whether the effect observed in our sample was genuine? – We don’t • Instead we use p values to indicate our level of certainty that our results represent a genuine effect present in the whole population P values • P values = the probability that the observed result was obtained by chance – i.e. when the null hypothesis is true • α level is set a priori (Usually 0.05) • If p < α level then we reject the null hypothesis and accept the experimental hypothesis – 95% certain that our experimental effect is genuine • If however, p > α level then we reject the experimental hypothesis and accept the null hypothesis Two types of errors • Type I error = false positive – α level of 0.05 means that there is 5% risk that a type I error will be encountered • Type II error = false negative t-tests • Compare two group means Hypothetical experiment Time Q – does viewing pictures of the Simpson and the Griffin family activate the same brain regions? Condition 1 = Simpson family faces Condition 2 = Griffin family faces Calculating T Difference between the means divided by the pooled standard error of the mean x1 x 2 t s x1 x2 2 Group 1 Group 2 s x1 x2 2 s1 s2 n1 n2 How do we apply this to fMRI data analysis? Time Degrees of freedom • = number of unconstrained data points • Which in this case = number of data points – 1. • Can use t value and df to find the associated p value • Then compare to the α level Different types of t-test • 2 sample t tests – Related = two samples related, i.e. same people in both conditions – Independent = two independent samples, i.e. diff people in 2 conditions • One sample t tests – compare the mean of one sample to a given value Another approach to group differences • Analysis Of VAriance (ANOVA) – Variances not means • Multiple groups e.g. Different facial expressions • H0 = no differences between groups • H1 = differences between groups Calculating F • F = the between group variance divided by the within group variance – the model variance/error variance • for F to be significant the between group variance should be considerably larger than the within group variance What can be concluded from a significant ANOVA? • There is a significant difference between the groups • NOT where this difference lies • Finding exactly where the differences lie requires further statistical analyses Different types of ANOVA • One-way ANOVA – One factor with more than 2 levels • Factorial ANOVAs – More than 1 factor • Mixed design ANOVAs – Some factors independent, others related Conclusions • T-tests assess if two group means differ significantly • Can compare two samples or one sample to a given value • ANOVAs compare more than two groups or more complicated scenarios • They use variances instead of means Further reading • Howell. Statistical methods for psychologists • Howitt and Cramer. An introduction to statistics in psychology • Huettel. Functional magnetic resonance imaging (especially chapter 12) Acknowledgements • MfD Slides 2005 – 2007 PART 2 • Correlation • Regression • Relevance to GLM and SPM Correlation • Strength and direction of the relationship between variables • Scattergrams Y Y Y X Positive correlation Y Y Y X Negative correlation No correlation • Describe correlation: covariance A statistic representing the degree to which 2 variables vary together – Covariance formula n cov( x, y ) ( x x)( y i 1 i i y) n n – cf. variance formula S x2 2 ( x x ) i i 1 n but… • the absolute value of cov(x,y) is also a function of the standard deviations of x and y. Describe correlation: Pearson correlation coefficient (r) • Equation cov( x, y) s = st dev of sample rxy sx s y – r = -1 (max. negative correlation); r = 0 (no constant relationship); r = 1 (max. positive correlation) • Limitations: 5 4 3 – Sensitive to extreme values, e.g. 2 1 0 0 1 2 3 4 – r is an estimate from the sample, but does it represent the population parameter? – Relationship not a prediction. 5 6 Summary • Correlation • Regression • Relevance to SPM Regression • Regression: Prediction of one variable from knowledge of one or more other variables. • Regression v. correlation: Regression allows you to predict one variable from the other (not just say if there is an association). • Linear regression aims to fit a straight line to data that for any value of x gives the best prediction of y. Best fit line, minimising sum of squared errors • Describing the line as in GCSE maths: y = m x + c • Here, ŷ = bx + a – ŷ : predicted value of y – b: slope of regression line – a: intercept ŷ = bx + a ε = ŷ, predicted = y i , observed ε = residual Residual error (ε): Difference between obtained and predicted values of y (i.e. y- ŷ). Best fit line (values of b and a) is the one that minimises the sum of squared errors (SSerror) (y- ŷ)2 • Minimise (y- ŷ)2 , which is (ybx+a)2 • Plotting SSerror for each possible regression line gives a parabola. • Minimum SSerror is at the bottom of the curve where the gradient is zero – and this can found with calculus. • Take partial derivatives of (ybx-a)2 and solve for 0 as simultaneous equations, giving: rs y a y bx b sx Sums of squared error (SSerror) How to minimise SSerror Gradient = 0 min SSerror Values of a and b How good is the model? • We can calculate the regression line for any data, but how well does it fit the data? • Total variance = predicted variance + error variance sy2 = sŷ2 + ser2 • Also, it can be shown that r2 is the proportion of the variance in y that is explained by our regression model r2 = sŷ2 / sy2 • Insert r2 sy2 into sy2 = sŷ2 + ser2 and rearrange to get: ser2 = 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: F(df ,df ) = ŷ er sŷ2 ser2 • And it follows that: r (n - 2) t(n-2) = √1 – r2 complicated rearranging r2 (n - 2)2 =......= 1 – r2 So all we need to know are r and n ! Summary • Correlation • Regression • Relevance to SPM General Linear Model • Linear regression is actually a form of the General Linear Model where the parameters are b, the slope of the line, and a, the intercept. y = bx + a +ε • A General Linear Model is just any model that describes the data in terms of a straight line One voxel: The GLM Our aim: Solve equation for β – tells us how much BOLD signal is explained by X a m b3 b4 b5 = b6 + b7 b8 b9 Y = X × b + e 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 = b0 + b1x1+ b2x2 +…..+ bnxn + 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 • This is what SPM does and will be explained soon… Summary • Correlation • Regression • Relevance to SPM Thanks!