T Tests and ANovas

1. T Tests and ANovas Jennifer Siegel

2. Objectives Statistical background Z-Test T-Test Anovas

3. Predicting the Future from a Sample • Science tries to predict the future • Genuine effect? • Attempt to strengthen predictions with stats • Use P-Value to indicate our level of certainty that result = genuine effect on whole population (more on this later…)

4. Normal Distribution

5. The Basics • Develop an experimental hypothesis • H0 = null hypothesis • H1 = alternative hypothesis • Statistically significant result • P Value = .05

6. P-Value • Probability that observed result is true • Level = .05 or 5% • 95% certain our experimental effect is genuine

7. Errors! • Type 1 = false positive • Type 2 = false negative • P = 1 – Probability of Type 1 error

9. Research Question Example • Let’s pretend you came up with the following theory… Having a baby increases brain volume (associated with possible structural changes)

10. Populations versus Samples Z - test T - test

11. Z-Test • Population

12. Some Problems with a Population-Based Study • Cost • Not able to include everyone • Too time consuming • Ethical right to privacy Realistically researchers can only do sample based studies

13. T-Test • T = differences between sample means / standard error of sample means • Degrees of freedom = sample size - 1

14. Two Sampled T-Tests: Pre and Post

15. Hypothesise • H0 = There is no difference in brain size before or after giving birth • H1 = The brain is significantly smaller or significantly larger after giving birth (difference detected)

16. Absolute Brain Volumes cm3 T=(1271-1236)/(119-113)

17. Results: p=.003 Women have a significantly larger brain after giving birth http://www.danielsoper.com/statcalc/calc08.aspx

18. Types of T-Tests One-sample (sample vs. hypothesized mean) Independent groups (2 separate groups) Repeated measures (same group, different measure)

19. More than 1 group???

20. ANOVA • ANalysis Of VAriance • Factor = what is being compared (type of pregnancy) • Levels = different elements of a factor (age of mother) • F-Statistic • Post hoc testing

21. Different types of Anova • 1 Way Anova • 1 factor with more than 2 levels • Factorial Anova • More than 1 factor • Mixed Design Anovas • Some factors are independent, others are related

22. What can be concluded from ANOVA • There is a significant difference somewhere between groups • NOT where the difference lies • Finding exactly where the difference lies requires further statistical analysis = post hoc analysis

24. Conclusions • Z-Tests for populations • T-Tests for samples • ANOVAS compare more than 2 groups in more complicated scenarios

25. Correlation and Linear Regression VarunV.Sethi

26. Objective Correlation Linear Regression Take Home Points.

27. With a few exceptions, every analysis is a variant of GLM

28. Correlation - How much linear is the relationship of two variables? (descriptive) Regression - How good is a linear model to explain my data? (inferential)

29. Correlation

30. Correlation Correlation reflects the noisiness and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom).

31. Y Y Y Y Y Y X X Positive correlation Negative correlation No correlation Correlation • Strength and direction of the relationship between variables • Scattergrams

33. Measures of Correlation • Covariance • 2) Pearson Correlation Coefficient (r)

34. 1) Covariance • The covariance is a statistic representing the degree to which 2 variables vary together • {Note that Sx2 = cov(x,x) }

35. A statistic representing the degree to which 2 variables vary together • Covariance formula • cf. variance formula

36. 2) Pearson correlation coefficient (r) • r is a kind of ‘normalised’ (dimensionless) covariance • r takes values fom -1 (perfect negative correlation) to 1 (perfect positive correlation). r=0 means no correlation (S = st dev of sample)

37. Pearson – ‘Strength of Linear Relation’ r = 0.816

38. Limitations: • Sensitive to extreme values • Relationship not a prediction. • Not Causality

39. Linear Regression

40. Regression: Prediction of one variable from knowledge of one or more other variables

41. How good is a linear model (y=ax+b) to explain the relationship of two variables? • If there is such a relationship, we can ‘predict’ the value y for a given x. (25, 7.498)

42. Linear dependence between 2 variables Two variables are linearly dependent when the increase of one variable is proportional to the increase of the other one y x Samples: - Energy needed to boil water - Money needed to buy coffeepots

44. εi = ŷi, predicted = yi , observed εi = residual Fiting data to a straight line (o viceversa): • Here, ŷ = ax + b • ŷ : predicted value of y • a: slope of regression line • b: intercept ŷ = ax + b • Residual error (εi): Difference between obtained and predicted values of y (i.e. yi- ŷi) • Best fit line (values of b and a) is the one that minimises the sum of squared errors (SSerror) (yi- ŷi)2

46. Adjusting the straight line to data: • Minimise (yi- ŷi)2 , which is (yi-axi+b)2 • Minimum SSerror is at the bottom of the curve where the gradient is zero – and this can found with calculus • Take partial derivatives of (yi-axi-b)2 respect parametres a and b and solve for 0 as simultaneous equations, giving: • This can always be done

47. 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

48. Is the model significant? • Do we get a significantly better prediction of y from our regression equation than by just predicting the mean? F-statistic

49. Practical Uses of Linear Regression • Prediction / Forecasting • Quantify strength between y and Xj( X1, X2, X3 )

50. General Linear Model • A General Linear Model is just any model that describes the data in terms of a straight line • 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