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Research Methods: 1 M.Sc. Physiotherapy/Podiatry/Pain Correlation, Regression and Basic Probability Relationships Between Variables Exploring relationships between variables What happens to one variable as another changes Relationships Between Variables • Correlation;the strength of the linear relationship between two variables. • Regression; the nature of that relationship, in terms of a mathematical equation. • In this module we are only concerned with linear relationships between variables. Correlation 12 Weight Relationship between Height and Weight 11 10 9 8 7 150 155 160 165 170 175 Height180 Correlation Correlation Coefficient = r -1 r 1 -1 perfect negative relationship 0 +1 perfect positive relationship Correlation; r = +1, perfect positive correlation Correlation; r = -1, perfect negative correlation Correlation; 0 < r < 1, positive correlation Correlation; -1 < r < 0, negative correlation Correlation; r 0, no linear relationship Correlation; r 0, no linear relationship Correlation • Closer to 1 the stronger • Relationships do not necessarily mean what you think, i.e. non-causal relationships Spurious Correlation • Coincidental Correlation; chance relationships • Indirect Correlation; related through some third variable • . Putting a value on the Linear Relationship • Pearson’s Product Moment Correlation Coefficient (PPM) • Parametric data - Quantitative data where it can be assumed both variables are normally distributed, = r Putting a value on the Linear Relationship • Spearmans Rank Correlation Coefficient • Non parametric - Ordinal data or quantitative data where one (or both) variables are not normally distributed. Calculated from the ranked data = (rho) Regression • Identify the nature of the relationship • Predict one variable from the other • The independent variable (plotted on the xaxis) determines the dependant variable (plotted on the y-axis) The Regression line The Method of Least Squares (the smallest sum of the squared distances) The Regression equation y = 0.836x + 2.800 Y = bX + a Y = the y-axis value X = the x-axis value b = the gradient (slope) of the line a = the intercept point with the y - axis The Regression prediction ? • Residuals • Coefficient of Determination • Coefficient of Determination * 100 = R squared (R2) • How good a fit the equation (and the line) is to the data Basic Probability Chance, possibilities and luck! • How likely is anything to happen, is one outcome more likely than any other? • What are the odds of one particular outcome occurring? • If you toss a dice what is the probability of getting a six ? Probability • • • • • p(Six) + p(Not Six) = 1 p(Six)1/6 + p(NotSix) 5/6 =1 If an event is certain to occur p=1 If an event is impossible p=0 All other events fall somewhere between 0 and 1, 0 < p < 1 Probability • p(Event) = trials of interest total number of trials • Theoretical and Relative frequencies Probability • 100 students attend a statistics lecture and 40 of them fall asleep within 10 minutes. • What is the probability that one of the students chosen at random will fall asleep within 10 minutes? • Pr (sleep) = 40/100 = 0.4 p = 0.4/40% Probability • 100 students attend a statistics lecture and 40 of them fall asleep within 10 minutes. • What is the probability that one of the students chosen at random will not fall asleep within 10 minutes? • Pr (not sleep) = 60/100 = 0.6 p = 0.6/60% Probability • • • • Complimentary rule of Probability Pr (not event) = 1 - Pr (event) Addition rule of Probability Pr(A or B) = Pr(A) + Pr(B) where A and B are mutually exclusive events