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Difference between standardized and unstandardized regression models

Uploaded: 3 years ago
Contributor: bio_man
Category: Statistics and Probability
Type: Lecture Notes
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Filename:   4552160.ppt (750 kB)
Page Count: 55
Credit Cost: 2
Views: 97
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Associations among Continuous Variables 3 characteristics of a relationship Direction Degree of association Form Regression & Correlation Correlation Correlation - Definition Correlation: a statistical technique that measures and describes the degree of linear relationship between two variables ’s r A value ranging from -1.00 to 1.00 indicating the strength and direction of the linear relationship. Absolute value indicates strength +/- indicates direction ’s r Deviation Score Formula Deviation Score Formula ’s r ’s r ’s r ’s r Z-score formula Z-score formula Z-score formula Hypothesis testing with r Hypotheses H0: ? = 0 HA : ? ? 0 Practice Practice Linear Regression Linear Regression But how do we describe the line? If two variables are linearly related it is possible to develop a simple equation to predict one variable from the other The outcome variable is designated the Y variable, and the predictor variable is designated the X variable E.g. centigrade to Fahrenheit: F = 32 + 1.8C this formula gives a specific straight line The Linear Equation The Linear Equation The Linear Equation The Linear Equation Slope and Intercept Equation of the line The slope b: the amount of change in y with one unit change in x The intercept a: the value of y when x is zero Slope and Intercept Equation of the line The slope The intercept When there is no linear association (r = 0), the regression line is horizontal. b=0. When the correlation is perfect (r = ± 1.00), all the points fall along a straight line with a slope When there is some linear association (0<|r|<1), the regression line fits as close to the points as possible and has a slope Where did this line come from? Regression lines Unstandardized Regression Line Equation of the line The slope The intercept Standardized Regression Line Equation of the line The slope The intercept Exercise: Revisit the seed data Calculate: r = b = a = ? = Write the regression equation: Write the standardized equation: Exercise: Revisit the seed data Calculate: r = .866 b = .375 a = 3.125 ? = .866 Write the regression equation: Write the standardized equation: Overview Correlation -Definition -Deviation Score Formula, Z score formula -Hypothesis Test Regression Intercept and Slope Unstandardized Regression Line Standardized Regression Line Hypothesis Tests Direction Positive(+) Negative (-) Degree of association Between –1 and 1 Absolute values signify strength Form Linear Non-linear Positive Large values of X = large values of Y, Small values of X = small values of Y. - e.g. IQ and SAT Large values of X = small values of Y Small values of X = large values of Y -e.g. SPEED and ACCURACY Negative Strong (tight cloud) Weak (diffuse cloud) Linear Non- linear What is the best fitting straight line? Regression Equation: Y = a + bX How closely are the points clustered around the line? ’s R Obs X Y A 1 1 B 1 3 C 3 2 D 4 5 E 6 4 F 7 5 Dataset X Y Scatterplot Below average on Y Below average on Y Above average on X Below average on X Above average on Y Above average on Y Above average on X Below average on X MEAN of Y MEAN of X The Logic of Correlation Cross-Product = For a strong positive association, the cross-products will mostly be positive Below average on Y Below average on Y Above average on X Below average on X Above average on Y Above average on Y Above average on X Below average on X MEAN of Y MEAN of X Cross-Product = For a strong negative association, the cross-products will mostly be negative The Logic of Correlation Below average on Y Below average on Y Above average on X Below average on X Above average on Y Above average on Y Above average on X Below average on X MEAN of Y MEAN of X Cross-Product = For a weak association, the cross-products will be mixed The Logic of Correlation SP (sum of products) = ? (X – X)(Y – Y) Deviation score formula SSY SSX SP 66.00 58.2 mean 84 74 E 72 64 D 70 59 C 63 56 B 41 38 A Humerus Femur SSY 1010 324 36 16 9 625 SSX 696.8 249.64 33.64 .64 4.84 408.04 SP 834 66.00 58.2 mean 284.4 18 15.8 84 74 E 34.8 6 5.8 72 64 D 3.2 4 0.8 70 59 C 6.6 -3 -2.2 63 56 B 505 -25 -20.2 41 38 A Humerus Femur = .99 For a strong positive association, the SP will be a big positive number SP (sum of products) = ? (X – X)(Y – Y) Deviation score formula Below average on Y Below average on Y Above average on X Below average on X Above average on Y Below Average on Y Above average on X Below average on X SP (sum of products) = ? (X – X)(Y – Y) Deviation score formula For a strong negative association, the SP will be a big negative number Below average on Y Below average on Y Above average on X Below average on X Above average on Y Below Average on Y Above average on X Below average on X SP (sum of products) = ? (X – X)(Y – Y) Deviation score formula For a weak association, the SP will be a small number (+ and – will cancel each other out) Below average on Y Below average on Y Above average on X Below average on X Above average on Y Below Average on Y Above average on X Below average on X Z score formula Standardized cross-products 15.89 13.20 s 66.00 58.2 mean 84 74 E 72 64 D 70 59 C 63 56 B 41 38 A ZXZY ZY ZX Humerus Femur 15.89 13.20 s 66.00 58.2 mean 1.133 1.197 84 74 E 0.378 0.439 72 64 D 0.252 0.061 70 59 C -0.189 -0.167 63 56 B -1.573 -1.530 41 38 A ZXZY ZY ZX Humerus Femur 15.89 13.20 s ?=3.976 66.00 58.2 mean 1.356 1.133 1.197 84 74 E 0.166 0.378 0.439 72 64 D 0.015 0.252 0.061 70 59 C 0.031 -0.189 -0.167 63 56 B 2.408 -1.573 -1.530 41 38 A ZXZY ZY ZX Humerus Femur r = .99 Formulas for R Z score formula Deviations formula Interpretation of R A measure of strength of association: how closely do the points cluster around a line? A measure of the direction of association: is it positive or negative? Interpretation of R r = .10 very small association, not usually reliable r = .20 small association r = .30 typical size for personality and social studies r = .40 moderate association r = .60 you are a research rock star r = .80 hmm, are you for real? Interpretation of R-squared The amount of covariation compared to the amount of total variation “The percent of total variance that is shared variance” E.g. “If r = .80, then X explains 64% of the variability in Y” (and vice versa) Test statistic = r Or just use table E.2 to find critical values of r SSY SSX SP mean 3.47 5.63 E 3.34 4.89 D 3.77 6.19 C 3.76 6.13 B 4.03 6.47 A tobacco alcohol SSY .30 SSX 1.55 SP .64 mean 3.47 5.63 E 3.34 4.89 D 3.77 6.19 C 3.76 6.13 B 4.03 6.47 A tobacco alcohol Properties of R A standardized statistic – will not change if you change the units of X or Y. (bc based on z-scores) The same whether X is correlated with Y or vice versa Fairly unstable with small n Vulnerable to outliers Has a skewed distribution F = 32 + 1.8(C) General form is Y = a + bX The prediction equation: Y’ = a+ bX Where a = intercept b = slope X = the predictor Y = the criterion a and b are constants in a given line; X and Y change F = 32 + 1.8(C) General form is Y = a + bX The prediction equation: Y’ = a + bX Where a = intercept b = slope X = the predictor Y = the criterion When b changes… F = 32 + 1.8(C) General form is Y = a + bX The prediction equation: Y’ = a + bX Where a = intercept b = slope X = the predictor Y = the criterion When a changes… F = 32 + 1.8(C) General form is Y = a + bX The prediction equation: Y’ = a + bX Where a = intercept b = slope X = the predictor Y = the criterion When both a and b change… The slope is influenced by r, but is not the same as r and our best estimate of age is 29.5 at all heights. It is a straight line which is drawn through a scatterplot, to summarize the relationship between X and Y It is the line that minimizes the squared deviations (Y’ – Y)2 We call these vertical deviations “residuals” Minimizing the squared vertical distances, or “residuals” Properties of b (slope) An unstandardized statistic – will change if you change the units of X or Y. Depends on whether Y is regressed on X or vice versa A person 1 stdev above the mean on height would be how many stdevs above the mean on weight? Properties of ? (standardized slope) An standardized statistic – will not change if you change the units of X or Y. Is equal to r, in simple linear regression 1.291 1.118 7 9 0.645 1.118 6 9 0.645 0 6 5 -0.645 0 4 5 -0.645 -1.118 4 1 -1.291 -1.118 3 1 ZD ZM RawD RawM 1.291 1.118 7 9 0.645 1.118 6 9 0.645 0 6 5 -0.645 0 4 5 -0.645 -1.118 4 1 -1.291 -1.118 3 1 ZD ZM RawD RawM Regression Coefficients Table sig t SEb b Variable X - SEa a Intercept Standardized Coefficient Standard error Unstandardized Coefficient Predictor Summary Correlation: ’s r Unstandardized Regression Line Standardized Regression Line Associations among Continuous Variables 3 characteristics of a relationship Direction Degree of association Form Regression & Correlation Correlation Correlation - Definition Correlation: a statistical technique that measures and describes the degree of linear relationship between two variables ’s r A value ranging from -1.00 to 1.00 indicating the strength and direction of the linear relationship. Absolute value indicates strength +/- indicates direction ’s r Deviation Score Formula Deviation Score Formula ’s r ’s r ’s r ’s r Z-score formula Z-score formula Z-score formula Hypothesis testing with r Hypotheses H0: ? = 0 HA : ? ? 0 Practice Practice Linear Regression Linear Regression But how do we describe the line? If two variables are linearly related it is possible to develop a simple equation to predict one variable from the other The outcome variable is designated the Y variable, and the predictor variable is designated the X variable E.g. centigrade to Fahrenheit: F = 32 + 1.8C this formula gives a specific straight line The Linear Equation The Linear Equation The Linear Equation The Linear Equation Slope and Intercept Equation of the line The slope b: the amount of change in y with one unit change in x The intercept a: the value of y when x is zero Slope and Intercept Equation of the line The slope The intercept When there is no linear association (r = 0), the regression line is horizontal. b=0. When the correlation is perfect (r = ± 1.00), all the points fall along a straight line with a slope When there is some linear association (0<|r|<1), the regression line fits as close to the points as possible and has a slope Where did this line come from? Regression lines Unstandardized Regression Line Equation of the line The slope The intercept Standardized Regression Line Equation of the line The slope The intercept Exercise Calculate: r = b = a = ? = Write the regression equation: Write the standardized equation: Exercise Calculate: r = .866 b = .375 a = 3.125 ? = .866 Write the regression equation: Write the standardized equation: Exercise in Excel Calculate: r = b = a = ? = Write the regression equation: Write the standardized equation: Sketch the scatterplot and regression line F = 32 + 1.8(C) General form is Y = a + bX The prediction equation: Y’ = a + bX Where a = intercept b = slope X = the predictor Y = the criterion Different b’s… F = 32 + 1.8(C) General form is Y = a + bX The prediction equation: Y’ = a + bX Where a = intercept b = slope X = the predictor Y = the criterion Different a’s… F = 32 + 1.8(C) General form is Y = a + bX The prediction equation: Y’ = a + bX Where a = intercept b = slope X = the predictor Y = the criterion Different a’s and b’s … Properties of ? (standardized slope) A standardized statistic – will not change if you change the units of X or Y. Is equal to r, in simple linear regression 7 9 6 9 6 5 4 5 4 1 3 1 Y X 7 9 6 9 6 5 4 5 4 1 3 1 Y X -6 9 -2 7 -4 4 -3 2 0 1 -1.5 1 Y X Associations among Continuous Variables 3 characteristics of a relationship Direction Degree of association Form Regression & Correlation Correlation Correlation - Definition Correlation: a statistical technique that measures and describes the degree of linear relationship between two variables ’s r A value ranging from -1.00 to 1.00 indicating the strength and direction of the linear relationship. Absolute value indicates strength +/- indicates direction ’s r Deviation Score Formula Deviation Score Formula ’s r ’s r ’s r ’s r Z-score formula Z-score formula Z-score formula Hypothesis testing with r Hypotheses H0: ? = 0 HA : ? ? 0 Practice Practice Linear Regression Linear Regression But how do we describe the line? If two variables are linearly related it is possible to develop a simple equation to predict one variable from the other The outcome variable is designated the Y variable, and the predictor variable is designated the X variable E.g. centigrade to Fahrenheit: F = 32 + 1.8C this formula gives a specific straight line The Linear Equation The Linear Equation The Linear Equation The Linear Equation Slope and Intercept Equation of the line The slope b: the amount of change in y with one unit change in x The intercept a: the value of y when x is zero Slope and Intercept Equation of the line The slope The intercept When there is no linear association (r = 0), the regression line is horizontal. b=0. When the correlation is perfect (r = ± 1.00), all the points fall along a straight line with a slope When there is some linear association (0<|r|<1), the regression line fits as close to the points as possible and has a slope Where did this line come from? Regression lines Unstandardized Regression Line Equation of the line The slope The intercept Standardized Regression Line Equation of the line The slope The intercept Exercise Calculate: r = b = a = ? = Write the regression equation: Write the standardized equation: Exercise Calculate: r = .866 b = .375 a = 3.125 ? = .866 Write the regression equation: Write the standardized equation: Exercise in Excel Calculate: r = b = a = ? = Write the regression equation: Write the standardized equation: Sketch the scatterplot and regression line

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