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39 Cards in this Set
- Front
- Back
Two of the most powerful and versatile approaches for investigating variable relationships
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correlation analysis
regression analysis |
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shows the relationship between two quantitative variables measured on the same individuals
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scatterplots
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scatter plot displays (3) of the relationship
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form (linear or nonlinear)
direction (+ or -) strength (no, weak, or strong) |
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relationship examined by our eyes may not be satisfactory in many cases we need numerical measurement to supplement graph
____is the measure we use |
correlation
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The correlation measures the
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direction and strength of the linear relationship between two quantitative variables
*Pearson correlation coefficient (r) |
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Pearson correlation coefficient is
standardized ______ |
covariance
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Covariance=
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degree to which X and Y vary together
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Cov(X,Y) > 0 means
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X and Y tend to move in the same direction
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Cov(X,Y) < 0 means
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X and Y tend to move in opposite directions
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Cov(X,Y) = 0 means
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X and Y are independent
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explain -1 ≤ r ≤ +1
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-closer to –1, the stronger the negative linear relationship
-the closer to 1, the stronger the positive linear relationship -the closer to 0, the weaker any linear relationship |
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T or F correlation is It is an indicator of causal relationship between variables
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F
its NOT |
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only legitimate way to try to establish a causal connection statistically is
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through the use of designed experiments
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what Statistical test can be used for the significance
of correlation coefficient if a single parameter |
t test
*we can use the original and convert it for correlation coefficient |
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linear regression:
Correlation treats two variables X and Y as_____ (it shows a ____relationship) |
equal
symmetric linear |
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In many cases, we want to study a ______linear relationship between X and Y.
meaning? |
-asymmetric
-One variable (X) influences (or predicts) the other variable (Y). X = IV or predictor variable Y = DV or outcome variable |
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Linear Regression Analysis=
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Describes how the DV (Y) changes as a single independent variable (X) changes (the effect of X on Y)
*called ‘simple’ linear regression analysis **summarizes the relationship between two variables if the form of the relationship is linear |
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linear regression model is used as a
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mathematical model to predict the value of DV (Y) based on a value of an IV (X)
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When a scatterplot displays a linear pattern, we can describe the overall pattern by
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drawing a straight line through the points.
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The equation of a line fitted to data gives a
compact description of the dependency of the___on the___ |
DV on the IV
*equation of a line is a mathematical model for the straight-line relationship |
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What is the “equation of line”?
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A straight line relating Y to X has an equation of the
form: Ŷ = a + bx a = intercept (Mean value of DV when IV is zero) b= slope (Amount by which DV changes on average when IV changes by one unit) |
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simple regression model
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fits a straight line through the set of n points in such a way that makes the sum of squared residuals of the model (that is, vertical distances between the points of the data set and the fitted line) as small as possible.
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simple regression (II)
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..
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When we have a scatterplot with a linear relationship between the DV (Y) and a single IV (X), we are often interested in summarizing the overall pattern
this is done how |
drawing a line on the graph=regression line.
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regression line =
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straight line that describes how Y changes as X changes.
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How to determine the best regression line?
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-line that comes the closest to the data points in the
vertical direction. There are many ways to make this distance “as small as possible.” -use method of least squares |
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least-squares regression line of Y on X is the line that
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makes the sum of the squares of the vertical distances of the data points from the line as small as possible.
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Sum of the Squares of all vertical distances=
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SS (ERROR)
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in the least squares method which is determined/calc first a or b
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b then a
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-can we use statisitcal tests to test the strength of the
relationship between the two variables in simple linear regression (the effect of X on Y) -if so what are the hypothesis |
-yes
-H0 : β = 0 (β = population slope) There is no linear relationship between X and Y (no effect of X on Y) H1 : β ≠ 0 There is a linear relationship (an effect of X on Y) |
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Statistical test for the significance of slope
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-just use/edit the t statistic
-If the observed value of t is greater than a critical value of t with DF = N-2 and α = .05, we may reject the null hypothesis |
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to apply the t test for the slope, the following assumptions are required (2)
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Normal distributions
Independent observations |
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As in ANOVA, we can also divide the variance/variation in the DV (Y) into different parts resulting from different sources
-In regression analysis, the total variation in Y is partitioned into(2) |
SS(Regression): The variation in Y explained by the regression line
SS(Error): The variation in Y unexplained by the regression line (residuals). |
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simple regression:
the_____is used for testing |
F statistic
Ho : β = 0 H1 : β ≠ 0 If the observed F value is greater than a critical value of F with DF(Reg) and DF(E) at α= .05, we may reject H0. |
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Coefficient of Determination (R^2)=
used to range |
-SS(regression)/SS(T)
*Proportion of the total variation in Y accounted for by the regression model. -assess goodness-of-fit of the regression model -0-1 |
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The larger R^2, the more ____of DV explained
0 = 1= |
variance
No explanation at all Perfect explanation. |
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in simple regression, r=
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√R^2
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example:
r^2=.710 means what |
71% of the total variance of the DV is explained by the IV
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Note that in simple regression analysis, both____(2)
tests are used for testing the significance of the single slope, resulting in the same conclusion. |
F and t
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