Презентация Correlation Regression онлайн

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    1,2,3,4,5,6,7,8,9,10,11
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Слайды и текст к этой презентации:

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The Question Are two
Содержание слайда: 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?
№14 слайд
Scatterplots Graphically
Содержание слайда: Scatterplots Graphically depicts the relationship between two variables in two dimensional space.
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Direct Relationship
Содержание слайда: Direct Relationship
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Inverse Relationship
Содержание слайда: Inverse Relationship
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An Example Does smoking
Содержание слайда: 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
№18 слайд
Trend?
Содержание слайда: Trend?
№19 слайд
Smoking and BP Note
Содержание слайда: 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?
№20 слайд
Heart Disease and Cigarettes
Содержание слайда: 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
Содержание слайда: The Data
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Scatterplot of Heart Disease
Содержание слайда: 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
Содержание слайда: 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
№25 слайд
Correlation Co-relation The
Содержание слайда: 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
Содержание слайда: 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
№27 слайд
Correlation Coefficient A
Содержание слайда: 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
Содержание слайда: Covariance The formula for co-variance is: How this works, and why? When would covXY be large and positive? Large and negative?
№30 слайд
Example
Содержание слайда: Example
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Example What the heck is a
Содержание слайда: Example What the heck is a covariance? I thought we were talking about correlation?
№32 слайд
Correlation Coefficient
Содержание слайда: Correlation Coefficient Pearson’s Product Moment Correlation Symbolized by r Covariance ÷ (product of the 2 SDs) Correlation is a standardized covariance
№33 слайд
Calculation for Example CovXY
Содержание слайда: Calculation for Example CovXY = 11.12 sX = 2.33 sY = 6.69
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Example Correlation . Sign is
Содержание слайда: Example Correlation = .713 Sign is positive Why? If sign were negative What would it mean? Would not change the degree of relationship.
№35 слайд
Factors Affecting r Range
Содержание слайда: 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.
№36 слайд
Factors Affecting r Outliers
Содержание слайда: Factors Affecting r Outliers Overestimate Correlation Underestimate Correlation
№37 слайд
Countries With Low
Содержание слайда: Countries With Low Consumptions
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Outliers
Содержание слайда: Outliers
№39 слайд
Testing Correlations So you
Содержание слайда: 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
Содержание слайда: „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
Содержание слайда: 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
Содержание слайда: 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 -
Содержание слайда: 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
Содержание слайда: 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
Содержание слайда: 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
Содержание слайда: The Data
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Regression Line Formula the
Содержание слайда: 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
Содержание слайда: 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
№51 слайд
Calculation Slope Intercept
Содержание слайда: Calculation Slope Intercept
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For Our Data CovXY . s X . .
Содержание слайда: 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
№53 слайд
Note The values we obtained
Содержание слайда: 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.
№54 слайд
Making a Prediction Second,
Содержание слайда: 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
№55 слайд
Accuracy of Prediction
Содержание слайда: 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 for
Содержание слайда: 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?
№58 слайд
Minimizing Residuals Again,
Содержание слайда: Minimizing Residuals Again, the problem lies with this definition of the mean: So, how do we get rid of the 0’s? Square them.
№59 слайд
Regression Line A
Содержание слайда: 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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