Statistics: Unlocking the Power of Data Lock 5 STAT 250 Dr. Kari Lock Morgan Simple Linear Regression SECTION 9.1 Inference for correlation Inference for slope Conditions for inference

Statistics: Unlocking the Power of Data Lock 5 Question of the Day Is the size of certain regions of your brain associated with the size of your social network?

Statistics: Unlocking the Power of Data Lock 5 Social Networks and the Brain Data from 40 students at City College London How to measure brain size? How to measure social network size? Source: R. Kanai, B. Bahrami, R. Roylance and G. Ree (2011). Online social network size is reflected in human brain structure, Proceedings of the Royal Society B: Biological Sciences. 10/19/11.Online social network size is reflected in human brain structure

Statistics: Unlocking the Power of Data Lock 5 Measuring Brain Size Structural Magnetic Resonance Imaging (MRI) Voxel-based morphometry (VBM) to compute regional grey matter volume based on T1-weighted anatomical MRI scans Brain regions found significant in initial study  Amygdala (emotion and emotional memory)  Middle temporal gyrus (social perception)  Entorhinal cortex (memory and navigation)  Superior temporal sulcus (perception of others) Response: normalized z-score of grey matter density for these brain regions

Statistics: Unlocking the Power of Data Lock 5 Brain Regions Image from Do our Brains Determine our Facebook Friend Count? ( our Brains Determine our Facebook Friend Count?

Statistics: Unlocking the Power of Data Lock 5 Social Networks and the Brain How to measure size of social network?  How many were present at your 18th or 21st birthday party?  If you were going to have a party now, how many people would you invite?  What is the total number of friends in your phonebook?  Write down the names of the people to whom you would send a text message marking a celebratory event. How many people is that?  Write down the names of people in your phonebook you would meet for a chat in a small group (one to three people). How many people is that?  How many friends have you kept from school and university whom you could have a friendly conversation with now?  How many friends do you have on ‘Facebook’?  How many friends do you have from outside school or university?  Write down the names of the people of whom you feel you could ask a favor and expect to have it granted. How many people is that?

Statistics: Unlocking the Power of Data Lock 5 Social Networks and the Brain r = Is the association significant?

Statistics: Unlocking the Power of Data Lock 5 ParameterDistributionStandard Error Proportion Normal Difference in Proportions Normal Meant, df = n – 1 Difference in Meanst, df = min(n 1, n 2 ) – 1 Correlationt, df = n – 2 Standard Error Formulas

Statistics: Unlocking the Power of Data Lock 5 Social Networks and the Brain Is the grey matter volume of these regions of the brain significantly correlated with number of Facebook friends? From n = 40 people, we find r =.436. Is this significant?

Statistics: Unlocking the Power of Data Lock 5 Social Networks and the Brain This provides strong evidence that the grey matter density of these regions of the brain and number of Facebook friends are positively correlated. 1. State hypotheses: 2. Check conditions: 3. Calculate test statistic: 4. Compute p-value: 5. Interpret in context: p-value =

Statistics: Unlocking the Power of Data Lock 5 Social Networks and the Brain Should you go out and add more Facebook friends to increase the size of your brain? a) Yes b) No

Statistics: Unlocking the Power of Data Lock 5 R2R2 R 2 is the proportion of the variability in the response variable, Y, that is explained by the explanatory variable, X For simple linear regression, R 2 = r 2 (R 2 is just the sample correlation squared)

Statistics: Unlocking the Power of Data Lock 5 R2R2 How much does the variability in Y decrease if you know X?

Statistics: Unlocking the Power of Data Lock 5 Regression in Minitab Stat -> Regression -> Fitted Line Plot = 0.19

Statistics: Unlocking the Power of Data Lock 5 Prediction Should you use this equation to predict the normalized size of these regions of your brain? a) Yes b) No

Statistics: Unlocking the Power of Data Lock 5 Sample to Population Everything we have done so far with regression is based solely on sample data Now, we will extend from the sample to the population Statistical inference!

Statistics: Unlocking the Power of Data Lock 5 Intercept Slope Simple Linear Model Random error

Statistics: Unlocking the Power of Data Lock 5 Inference for the Slope Test for whether the slope is significantly different from 0 (whether there is any linear relationship between x and y): Confidence interval for the true slope

Statistics: Unlocking the Power of Data Lock 5 Inference for the Slope

Statistics: Unlocking the Power of Data Lock 5 Regression in Minitab Stat -> Regression -> Regression -> Fit Regression Model

Statistics: Unlocking the Power of Data Lock 5 Inference for Slope Is the slope significantly different from 0? (a) Yes (b) No n = 40 Give a 95% confidence interval for the true slope.

Statistics: Unlocking the Power of Data Lock 5 Hypothesis Test

Statistics: Unlocking the Power of Data Lock 5 Regression in Minitab Stat -> Regression -> Regression -> Fit Regression Model

Statistics: Unlocking the Power of Data Lock 5 Two Quantitative Variables The t-statistic (and p-value) for a test for a non-zero slope and a test for a non-zero correlation are identical! They are equivalent ways of testing for a linear association between two quantitative variables.

Statistics: Unlocking the Power of Data Lock 5 Confidence Interval

Statistics: Unlocking the Power of Data Lock 5 Multiple Testing?

Statistics: Unlocking the Power of Data Lock 5 False Positive (Type I Error) Protection To further protect against Type I errors, they performed two independent analysis on two separate samples (n = 125, then n = 40)

Statistics: Unlocking the Power of Data Lock 5 Real-World Network Size What about real-world network size?

Statistics: Unlocking the Power of Data Lock 5 Inference based on the simple linear model is only valid if the following conditions hold: 1)Linearity 2)Constant Variability of Residuals 3)Normality of Residuals 4)Independence Conditions

Statistics: Unlocking the Power of Data Lock 5 The relationship between x and y is linear (it makes sense to draw a line through the scatterplot) Linearity

Statistics: Unlocking the Power of Data Lock 5 Dog Years 1 dog year = 7 human years Linear: human age = 7×dog age Charlie From “The old rule-of-thumb that one dog year equals seven years of a human life is not accurate. The ratio is higher with youth and decreases a bit as the dog ages.” LINEAR ACTUAL A linear model can still be useful, even if it doesn’t perfectly fit the data.

Statistics: Unlocking the Power of Data Lock 5 “All models are wrong, but some are useful” -George Box

Statistics: Unlocking the Power of Data Lock 5 Residuals (errors) The errors are normally distributed The average of the errors is 0 The standard deviation of the errors is constant for all cases Conditions for residuals: Check with a histogram (Always true for least squares regression) Constant spread of points around the line

Statistics: Unlocking the Power of Data Lock 5 Regression in Minitab Is the association approximately linear? a)Yes b)No Is the spread of the points around the line approximately constant? a)Yes b)No

Statistics: Unlocking the Power of Data Lock 5 Histogram of Residuals Are the residuals approximately normally distributed? a)Yes b)No

Statistics: Unlocking the Power of Data Lock 5 Non-Constant Variability

Statistics: Unlocking the Power of Data Lock 5 Non-Normal Residuals

Statistics: Unlocking the Power of Data Lock 5 Cases must be independent of each other (one case’s values does not affect another case’s values) Most common violation of this: data over time What would make the independence condition satisfied or violated in the social network and brain size data? Independence

Statistics: Unlocking the Power of Data Lock 5 If the association isn’t linear: don’t use simple linear regression If variability is not constant, residuals are not normal, or cases not independent: The model itself is still valid, but inference may not be accurate If you want to do something more fancy so the conditions are met… take STAT 462! Conditions not Met?

Statistics: Unlocking the Power of Data Lock 5 1) Plot your data! Association approximately linear? Outliers? Constant variability? 2) Fit the model (least squares) 3) Use the model Interpret coefficients Make predictions 4) Look at histogram of residuals (normal?) 5) Inference (extend to population) Inference on slope (interval and test) Simple Linear Regression

Statistics: Unlocking the Power of Data Lock 5 To Do Read Section 9.1 Do HW 9.1 (due Friday, 12/4)