🗊Презентация Correlation Regression

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Correlation Regression, слайд №1
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The Question Are two variables related? Does one increase as the other increases? e. g. skills and income Does one decrease as the other increases? e. g. health problems and nutrition How can we get a numerical measure of the degree of relationship?
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The Question Are two variables related? Does one increase as the other increases? e. g. skills and income Does one decrease as the other increases? e. g. health problems and nutrition How can we get a numerical measure of the degree of relationship?

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Scatterplots Graphically depicts the relationship between two variables in two dimensional space.
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Scatterplots Graphically depicts the relationship between two variables in two dimensional space.

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Direct Relationship
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Direct Relationship

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Inverse Relationship
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Inverse Relationship

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An Example Does smoking cigarettes increase systolic blood pressure? Plotting number of cigarettes smoked per day against systolic blood pressure Fairly moderate relationship Relationship is positive
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An Example Does smoking cigarettes increase systolic blood pressure? Plotting number of cigarettes smoked per day against systolic blood pressure Fairly moderate relationship Relationship is positive

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Trend?
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Trend?

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Smoking and BP Note relationship is moderate, but real. Why do we care about relationship? What would conclude if there were no relationship? What if the relationship were near perfect? What if the relationship were negative?
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Smoking and BP Note relationship is moderate, but real. Why do we care about relationship? What would conclude if there were no relationship? What if the relationship were near perfect? What if the relationship were negative?

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Heart Disease and Cigarettes Data on heart disease and cigarette smoking in 21 developed countries Data have been rounded for computational convenience. The results were not affected.
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Heart Disease and Cigarettes Data on heart disease and cigarette smoking in 21 developed countries Data have been rounded for computational convenience. The results were not affected.

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The Data
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The Data

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Scatterplot of Heart Disease CHD Mortality goes on Y axis Why? Cigarette consumption on X axis Why? What does each dot represent? Best fitting line included for clarity
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Scatterplot of Heart Disease CHD Mortality goes on Y axis Why? Cigarette consumption on X axis Why? What does each dot represent? Best fitting line included for clarity

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What Does the Scatterplot Show? As smoking increases, so does coronary heart disease mortality. Relationship looks strong Not all data points on line. This gives us “residuals” or “errors of prediction” To be discussed later
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What Does the Scatterplot Show? As smoking increases, so does coronary heart disease mortality. Relationship looks strong Not all data points on line. This gives us “residuals” or “errors of prediction” To be discussed later

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Correlation Co-relation The relationship between two variables Measured with a correlation coefficient Most popularly seen correlation coefficient: Pearson Product-Moment Correlation
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Correlation Co-relation The relationship between two variables Measured with a correlation coefficient Most popularly seen correlation coefficient: Pearson Product-Moment Correlation

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Types of Correlation Positive correlation High values of X tend to be associated with high values of Y. As X increases, Y increases Negative correlation High values of X tend to be associated with low values of Y. As X increases, Y decreases No correlation No consistent tendency for values on Y to increase or decrease as X increases
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Types of Correlation Positive correlation High values of X tend to be associated with high values of Y. As X increases, Y increases Negative correlation High values of X tend to be associated with low values of Y. As X increases, Y decreases No correlation No consistent tendency for values on Y to increase or decrease as X increases

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Correlation Coefficient A measure of degree of relationship. Between 1 and -1 Sign refers to direction. Based on covariance Measure of degree to which large scores on X go with large scores on Y, and small scores on X go with small scores on Y
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Correlation Coefficient A measure of degree of relationship. Between 1 and -1 Sign refers to direction. Based on covariance Measure of degree to which large scores on X go with large scores on Y, and small scores on X go with small scores on Y

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Covariance The formula for co-variance is: How this works, and why? When would covXY be large and positive? Large and negative?
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Covariance The formula for co-variance is: How this works, and why? When would covXY be large and positive? Large and negative?

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Example
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Example

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Example What the heck is a covariance? I thought we were talking about correlation?
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Example What the heck is a covariance? I thought we were talking about correlation?

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Correlation Coefficient Pearson’s Product Moment Correlation Symbolized by r Covariance ÷ (product of the 2 SDs) Correlation is a standardized covariance
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Correlation Coefficient Pearson’s Product Moment Correlation Symbolized by r Covariance ÷ (product of the 2 SDs) Correlation is a standardized covariance

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Calculation for Example CovXY = 11.12 sX = 2.33 sY = 6.69
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Calculation for Example CovXY = 11.12 sX = 2.33 sY = 6.69

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Example Correlation = .713 Sign is positive Why? If sign were negative What would it mean? Would not change the degree of relationship.
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Example Correlation = .713 Sign is positive Why? If sign were negative What would it mean? Would not change the degree of relationship.

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Factors Affecting r Range restrictions Looking at only a small portion of the total scatter plot (looking at a smaller portion of the scores’ variability) decreases r. Reducing variability reduces r Nonlinearity The Pearson r measures the degree of linear relationship between two variables If a strong non-linear relationship exists, r will provide a low, or at least inaccurate measure of the true relationship.
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Factors Affecting r Range restrictions Looking at only a small portion of the total scatter plot (looking at a smaller portion of the scores’ variability) decreases r. Reducing variability reduces r Nonlinearity The Pearson r measures the degree of linear relationship between two variables If a strong non-linear relationship exists, r will provide a low, or at least inaccurate measure of the true relationship.

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Factors Affecting r Outliers Overestimate Correlation Underestimate Correlation
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Factors Affecting r Outliers Overestimate Correlation Underestimate Correlation

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Countries With Low Consumptions
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Countries With Low Consumptions

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Outliers
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Outliers

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Testing Correlations So you have a correlation. Now what? In terms of magnitude, how big is big? Small correlations in large samples are “big.” Large correlations in small samples aren’t always “big.” Depends upon the magnitude of the correlation coefficient AND The size of your sample.
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Testing Correlations So you have a correlation. Now what? In terms of magnitude, how big is big? Small correlations in large samples are “big.” Large correlations in small samples aren’t always “big.” Depends upon the magnitude of the correlation coefficient AND The size of your sample.

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„Regression” refers to the process of fitting a simple line to datapoints, Historically, linear regression was first used to explain the height of men by the height of their fathers. „Regression” refers to the process of fitting a simple line to datapoints, Historically, linear regression was first used to explain the height of men by the height of their fathers.
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„Regression” refers to the process of fitting a simple line to datapoints, Historically, linear regression was first used to explain the height of men by the height of their fathers. „Regression” refers to the process of fitting a simple line to datapoints, Historically, linear regression was first used to explain the height of men by the height of their fathers.

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What is regression? How do we predict one variable from another? How does one variable change as the other changes? Influence
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What is regression? How do we predict one variable from another? How does one variable change as the other changes? Influence

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Linear Regression A technique we use to predict the most likely score on one variable from those on another variable Uses the nature of the relationship (i.e. correlation) between two variables to enhance your prediction
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Linear Regression A technique we use to predict the most likely score on one variable from those on another variable Uses the nature of the relationship (i.e. correlation) between two variables to enhance your prediction

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Linear Regression: Parts Y - the variables you are predicting i.e. dependent variable X - the variables you are using to predict i.e. independent variable - your predictions (also known as Y’)
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Linear Regression: Parts Y - the variables you are predicting i.e. dependent variable X - the variables you are using to predict i.e. independent variable - your predictions (also known as Y’)

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Why Do We Care? We may want to make a prediction. More likely, we want to understand the relationship. How fast does CHD mortality rise with a one unit increase in smoking? Note: we speak about predicting, but often don’t actually predict.
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Why Do We Care? We may want to make a prediction. More likely, we want to understand the relationship. How fast does CHD mortality rise with a one unit increase in smoking? Note: we speak about predicting, but often don’t actually predict.

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An Example Cigarettes and CHD Mortality again Data repeated on next slide We want to predict level of CHD mortality in a country averaging 10 cigarettes per day.
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An Example Cigarettes and CHD Mortality again Data repeated on next slide We want to predict level of CHD mortality in a country averaging 10 cigarettes per day.

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The Data
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The Data

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Regression Line Formula = the predicted value of Y (e.g. CHD mortality) X = the predictor variable (e.g. average cig./adult/country)
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Regression Line Formula = the predicted value of Y (e.g. CHD mortality) X = the predictor variable (e.g. average cig./adult/country)

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Regression Coefficients “Coefficients” are a and b b = slope Change in predicted Y for one unit change in X a = intercept value of when X = 0
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Regression Coefficients “Coefficients” are a and b b = slope Change in predicted Y for one unit change in X a = intercept value of when X = 0

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Calculation Slope Intercept
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Calculation Slope Intercept

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For Our Data CovXY = 11.12 s2X = 2.332 = 5.447 b = 11.12/5.447 = 2.042 a = 14.524 - 2.042*5.952 = 2.32
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For Our Data CovXY = 11.12 s2X = 2.332 = 5.447 b = 11.12/5.447 = 2.042 a = 14.524 - 2.042*5.952 = 2.32

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Note: The values we obtained are shown on printout. The intercept is the value in the B column labeled “constant” The slope is the value in the B column labeled by name of predictor variable.
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Note: The values we obtained are shown on printout. The intercept is the value in the B column labeled “constant” The slope is the value in the B column labeled by name of predictor variable.

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Making a Prediction Second, once we know the relationship we can predict We predict 22.77 people/10,000 in a country with an average of 10 C/A/D will die of CHD
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Making a Prediction Second, once we know the relationship we can predict We predict 22.77 people/10,000 in a country with an average of 10 C/A/D will die of CHD

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Accuracy of Prediction Finnish smokers smoke 6 C/A/D We predict: They actually have 23 deaths/10,000 Our error (“residual”) = 23 - 14.619 = 8.38 a large error
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Accuracy of Prediction Finnish smokers smoke 6 C/A/D We predict: They actually have 23 deaths/10,000 Our error (“residual”) = 23 - 14.619 = 8.38 a large error

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Residuals When we predict Ŷ for a given X, we will sometimes be in error. Y – Ŷ for any X is a an error of estimate Also known as: a residual We want to Σ(Y- Ŷ) as small as possible. BUT, there are infinitely many lines that can do this. Just draw ANY line that goes through the mean of the X and Y values. Minimize Errors of Estimate… How?
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Residuals When we predict Ŷ for a given X, we will sometimes be in error. Y – Ŷ for any X is a an error of estimate Also known as: a residual We want to Σ(Y- Ŷ) as small as possible. BUT, there are infinitely many lines that can do this. Just draw ANY line that goes through the mean of the X and Y values. Minimize Errors of Estimate… How?

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Minimizing Residuals Again, the problem lies with this definition of the mean: So, how do we get rid of the 0’s? Square them.
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Minimizing Residuals Again, the problem lies with this definition of the mean: So, how do we get rid of the 0’s? Square them.

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Regression Line: A Mathematical Definition The regression line is the line which when drawn through your data set produces the smallest value of: Called the Sum of Squared Residual or SSresidual Regression line is also called a “least squares line.”
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Regression Line: A Mathematical Definition The regression line is the line which when drawn through your data set produces the smallest value of: Called the Sum of Squared Residual or SSresidual Regression line is also called a “least squares line.”

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