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Introduction to Statistics Steven A. Jones Biomedical Engineering Louisiana Tech University (Created for our NSF-funded Research Experiences in Micro/Nano Engineering Program) Experimental Design 1.Develop a Hypothesis 2.Design Statistical Analysis 3.Set up Experiment 4.Test Experiment (Positive Control) 5.Collect Data 6.Perform Statistical Analysis Step 2 must not be done after the data have been collected! Some Common Probability Distributions Uniform Distribution Probability Density Probability Density Gaussian Distribution 0.15 0.1 0.05 0 0 5 10 15 20 25 Probability Density Value of x 0.4 0.2 0.1 0 0 5 10 15 Value of x 20 25 Gaussian: Sum of numbers. Rayleigh Distribution 0.3 Uniform: e.g. Dice throw. 0.2 Rayleigh: Square root of the sum of the squares of two Gaussians. 0.1 0 0 2 4 Value of X 6 Introductory Question There is a 60% chance of rain on Friday and a 45% chance of rain on Saturday. What is the probability that it will rain on Friday and Saturday? Introductory Question There is a 60% chance of rain on Friday and a 45% chance of rain on Saturday. What is the probability that it will rain on either Friday or Saturday? Axioms of Probability For independent outcomes A and B: P A & B P APB P A 1 P A P A or B 1 PA PB 1 1 P A 1 PB P A PB P APB If probabilities are small, then they can be added when “or” is used. Relationships Among Probability Distributions Assume that xi are uniformly distributed. Then: N y i Ai xi Is Gaussian (Normal) distributed for N sufficiently large. i 1 z y12 y22 w z Is 2 distributed. Is Rayleigh distributed. Simple Distribution You expect that the correct value for a height measurement is 12 meters, and the standard deviation is 3 meters. One way to determine whether or not you are correct is to take some measurements. What could you conclude if you made one measurement and the value fell as follows on the expected distribution? Probability Density Gaussian Distribution 0.15 0.1 0.05 0 0 5 10 15 Value of x 20 25 What if three values fell as follows on the expected distribution? Probability Density Gaussian Distribution 0.15 0.1 0.05 0 0 5 10 15 Value of x 20 25 Did the two data sets to the left come from different distributions? Set 1: Set 2: 9 9.5 10 10.5 11 11.5 12 Value of X Set 1: What about these two? Set 2: 9 9.5 10 10.5 Value of X 11 11.5 12 How confident are you that the data sets in each plot below come from different distributions? 9 9.5 10 10.5 11 11.5 12 9 9.5 10 10.5 11 11.5 12 11.5 12 Value of X Value of X Lower standard deviation 9 9.5 10 10.5 11 11.5 Value of X Smaller difference in means 12 9 9.5 10 10.5 Value of X Fewer data points 11 Student’s T Test A Student’s T test measures the confidence you can have that two values are inherently different, based on three parameters 1. Difference of the means 2. Standard deviations 3. Number of data points obtained Particularly useful when there are multiple confounding variables. E.g. Blood pressure drugs – are we, on average, lowering blood pressure? A Student’s T test is used to answer the following question: Given: • Difference of the means • Standard deviations • Number of data points obtained • That these data come from normal distributions What is the probability (p) that they came from the same underlying distribution? Example Given the mean and standard deviation for pressure, along with the number of points measured from a clinical drug trial, what is the probability that the drug had an effect on the distribution (i.e. that it changed the blood pressure of these individuals on average). Sample Mean: Mean from the sample that was taken (the 2000 people in the drug trial). Distribution Mean: Mean that would occur if you could give the drug to everyone in the world and do the measurement. Underlying and Sample Distributions Uniform Distribution 30 Uniform (Theory) Gaussian Distribution 25 Gaussian (Theory) Frequency 20 15 10 5 0 -3 -2 -1 0 Bin 1 2 3 Statistical Tests You Should Know T-test: Are the means of two data sets the same? F-test: Are the standard deviations of two data sets the same? Chi-Squred Test: Does the distribution of a data set match a proposed distribution? Anova: Like an F-test for multiple variables. Pearson’s Correlation Coefficient: Does one variable depend on another? To Run a T-Test Calculate the mean of the data. Calculate the standard deviation of the data. Determine the T statistic (e.g. Nx ) From T determine p. p is “the probability that you would get a difference in means this large or smaller, given that the two measurement sets come from the same distribution.” Interpretation of T test You set the value that you consider significant. Medical applications: p < 0.05 is “significant.” Since p < 0.05 is a 1/20 probability, you will typically be wrong once in every 20 T tests. Hypothesis Null hypothesis: statement that the two distributions are the same. i.e. “Altase causes no change in blood pressure.” Alternative hypothesis: Can vary. Altase reduces the mean blood pressure. Altase changes the mean blood pressure. One-Tailed vs Two-Tailed Depends on “alternative hypothesis.” One tail: If alternative hypothesis is that one mean is greater than the other. Two tail: If alternative hypothesis is that the means are different. Saying that one of the means is greater is more restrictive. The confidence you have in your result depends on the prediction (1st law of the frisbee). Example A friend throws a frisbee. It bounces off a pole, goes to the roof of the house, rolls along an arc, flips off the gutter, and then lands in the fountain. Are you impressed? A friend predicts that the frisbee will do the above, and then it happens. Are you impressed? As with the frisbee, statistical analysis depends on how far you are willing to stick your neck out. Confidence Interval States the range of values that contains the true value within a given percent confidence. Depends on , number of samples, and desired confidence. Not a statistical test of significance, but related to the T-test • The more samples we have, the narrower the confidence interval. Probability Density Gaussian Distribution 0.15 0.1 0.05 0 0 5 10 15 Value of x 20 25