Regression & Correlation. Review: Types of Variables & Steps in Analysis.
-
Upload
willa-malone -
Category
Documents
-
view
221 -
download
2
Transcript of Regression & Correlation. Review: Types of Variables & Steps in Analysis.
![Page 1: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/1.jpg)
Regression & Correlation
![Page 2: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/2.jpg)
Review:Types of Variables
&Steps in Analysis
![Page 3: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/3.jpg)
Types of Variables
Nominal / Categorical: each value is distinct category [gender, blood type, city]
Scale / Interval: linear measure, same intervalbetween each value [age, weight, IQ, GPA, SAT, income]
Ordinal: ranking, un-equal intervalsbetween values
[Likert scale, preference ranking]
![Page 4: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/4.jpg)
Variables & Statistical TestsVariable Type Example Common Stat
MethodNominal by nominal
Blood type by gender
Chi-square
Scale by nominal GPA by gender
GPA by major
T-test
Analysis of Variance
Scale by scale Weight by height
GPA by SAT
Regression
Correlation
![Page 5: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/5.jpg)
Evaluating an hypothesis
• Step 1: What is the relationship in the sample?
• Step 2: How confidently can one generalize from the sample to the universe from which it comes?
![Page 6: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/6.jpg)
Evaluating an hypothesis
Relationship in Sample
Statistical Significance
2 ordinal vars. Cross-tab / contingency table
“p value” from Chi Square
Scale dep. & 2-cat indep.
Means for each category
“p value” from t-test
Scale dep. & 3+ cat indep.
Means for each category
“p value” from ANOVA
2 scale vars. Regression line
Correlation
“p value” from reg or correlation
![Page 7: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/7.jpg)
Relationships betweenScale Variables
Regression
Correlation
![Page 8: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/8.jpg)
Regression• Amount that a dependent variable
increases (or decreases) for each unit increase in an independent variable.
• Expressed as equation for a line –
y = m(x) + b – the “regression line”
• Interpret by slope of the line: m
(Or: interpret by “odds ratio” in “logistic regression”)
![Page 9: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/9.jpg)
Correlation• Strength of association of scale measures
• r = -1 to 0 to +1
+1 perfect positive correlation
-1 perfect negative correlation
0 no correlation
• Interpret r in terms of variance
![Page 10: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/10.jpg)
Mean&
Variance
![Page 11: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/11.jpg)
Survey of Classn = 42
• Height• Mother’s height• Mother’s education• SAT• Estimate IQ• Well-being
(7 pt. Likert)
• Weight• Father’s education• Family income• G.P.A.• Health (7 pt. Likert)
![Page 12: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/12.jpg)
Frequency Table for: HEIGHT Valid CumValue Label Value Frequency Percent Percent Percent 59.00 1 2.4 2.4 2.4 61.00 2 4.8 4.8 7.1 62.00 3 7.1 7.1 14.3 63.00 3 7.1 7.1 21.4 65.00 5 11.9 11.9 33.3 66.00 3 7.1 7.1 40.5 67.00 4 9.5 9.5 50.0 68.00 5 11.9 11.9 61.9 69.00 1 2.4 2.4 64.3 70.00 6 14.3 14.3 78.6 71.00 1 2.4 2.4 81.0 72.00 4 9.5 9.5 90.5 73.00 3 7.1 7.1 97.6 74.00 1 2.4 2.4 100.0 ------- ------- ------- Total 42 100.0 100.0 Valid cases 42 Missing cases 0
![Page 13: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/13.jpg)
Frequency Table for: HEIGHT Valid CumValue Label Value Frequency Percent Percent Percent 59.00 1 2.4 2.4 2.4 61.00 2 4.8 4.8 7.1 62.00 3 7.1 7.1 14.3 63.00 3 7.1 7.1 21.4 65.00 5 11.9 11.9 33.3 66.00 3 7.1 7.1 40.5 67.00 4 9.5 9.5 50.0 68.00 5 11.9 11.9 61.9 69.00 1 2.4 2.4 64.3 70.00 6 14.3 14.3 78.6 71.00 1 2.4 2.4 81.0 72.00 4 9.5 9.5 90.5 73.00 3 7.1 7.1 97.6 74.00 1 2.4 2.4 100.0 ------- ------- ------- Total 42 100.0 100.0 Valid cases 42 Missing cases 0 Descriptive Statistics for: HEIGHT ValidVariable Mean Std Dev Variance Range Minimum Maximum N HEIGHT 67.33 3.87 14.96 15.00 59.00 74.00 42
mean
![Page 14: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/14.jpg)
Variance
x i - Mean )2
Variance = s2 = ----------------------- N
Standard Deviation = s = variance
![Page 15: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/15.jpg)
Frequency Table for: WEIGHT Valid CumValue Label Value Frequency Percent Percent Percent 115.00 1 2.4 2.4 2.4 120.00 1 2.4 2.4 4.8 124.00 1 2.4 2.4 7.1 125.00 4 9.5 9.5 16.7 128.00 1 2.4 2.4 19.0 130.00 6 14.3 14.3 33.3 135.00 4 9.5 9.5 42.9 136.00 1 2.4 2.4 45.2 140.00 3 7.1 7.1 52.4 145.00 2 4.8 4.8 57.1 150.00 3 7.1 7.1 64.3 155.00 2 4.8 4.8 69.0 160.00 6 14.3 14.3 83.3 165.00 2 4.8 4.8 88.1 170.00 1 2.4 2.4 90.5 185.00 1 2.4 2.4 92.9 190.00 2 4.8 4.8 97.6 210.00 1 2.4 2.4 100.0 ------- ------- ------- Total 42 100.0 100.0 Valid cases 42 Missing cases 0 Descriptive Statistics for: WEIGHT ValidVariable Mean Std Dev Variance Range Minimum Maximum N WEIGHT 146.38 21.30 453.80 95.00 115.00 210.00 42
mean
![Page 16: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/16.jpg)
Relationship of weight & height:
Regression Analysis
![Page 17: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/17.jpg)
![Page 18: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/18.jpg)
“Least Squares” Regression Line
Dependent = ( B ) (Independent) + constant
weight = ( B ) ( height ) + constant
![Page 19: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/19.jpg)
Regression line
![Page 20: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/20.jpg)
Regression: WEIGHT on HEIGHT Multiple R .59254R Square .35110Adjusted R Square .33488Standard Error 17.37332 Analysis of Variance DF Sum of Squares Mean SquareRegression 1 6532.61322 6532.61322Residual 40 12073.29154 301.83229 F = 21.64319 Signif F = .0000 ------------------ Variables in the Equation ------------------ Variable B SE B Beta T Sig T HEIGHT 3.263587 .701511 .592541 4.652 .0000(Constant) -73.367236 47.311093 -1.551
[ Equation: Weight = 3.3 ( height ) - 73 ]
![Page 21: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/21.jpg)
Regression line
W = 3.3 H - 73
![Page 22: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/22.jpg)
Strength of Relationship
“Goodness of Fit”: Correlation
How well does the regression line “fit” the data?
![Page 23: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/23.jpg)
![Page 24: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/24.jpg)
Frequency Table for: WEIGHT Valid CumValue Label Value Frequency Percent Percent Percent 115.00 1 2.4 2.4 2.4 120.00 1 2.4 2.4 4.8 124.00 1 2.4 2.4 7.1 125.00 4 9.5 9.5 16.7 128.00 1 2.4 2.4 19.0 130.00 6 14.3 14.3 33.3 135.00 4 9.5 9.5 42.9 136.00 1 2.4 2.4 45.2 140.00 3 7.1 7.1 52.4 145.00 2 4.8 4.8 57.1 150.00 3 7.1 7.1 64.3 155.00 2 4.8 4.8 69.0 160.00 6 14.3 14.3 83.3 165.00 2 4.8 4.8 88.1 170.00 1 2.4 2.4 90.5 185.00 1 2.4 2.4 92.9 190.00 2 4.8 4.8 97.6 210.00 1 2.4 2.4 100.0 ------- ------- ------- Total 42 100.0 100.0 Valid cases 42 Missing cases 0 Descriptive Statistics for: WEIGHT ValidVariable Mean Std Dev Variance Range Minimum Maximum N
WEIGHT 146.38 21.30 453.80 95.00 115.00 210.00 42
mean
![Page 25: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/25.jpg)
mean
Variance = 454
![Page 26: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/26.jpg)
Regression line
mean
![Page 27: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/27.jpg)
Correlation: “Goodness of Fit”• Variance (average sum of squared
distances from mean) = 454
• “Least squares” (average sum of squared distances from regression line) = 295
• 454 – 295 = 159 159 / 454 = .35
• Variance is reduced 35% by calculating from regression line
![Page 28: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/28.jpg)
r2 = % of variance in WEIGHT
“explained” by HEIGHT
Correlation coefficient = r
![Page 29: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/29.jpg)
Correlation: HEIGHT with WEIGHT HEIGHT WEIGHT HEIGHT 1.0000 .5925 ( 42) ( 42) P= . P= .000 WEIGHT .5925 1.0000 ( 42) ( 42) P= .000 P= .
![Page 30: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/30.jpg)
r = .59
r2 = .35
HEIGHT “explains” 35% of variance in WEIGHT
![Page 31: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/31.jpg)
Sentence & G.P.A.
• Regression: form of relationship
• Correlation: strength of relationship
• p value: statistical significance
![Page 32: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/32.jpg)
grade point average
grade point average
4.003.903.803.753.703.403.333.203.00
Fre
qu
en
cy
7
6
5
4
3
2
1
0
Statistics
grade point average23
1
3.5752
.35057
.12290
Valid
Missing
N
Mean
Std. Deviation
Variance
G. P. A.
![Page 33: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/33.jpg)
jail sentence
jail sentence
18.0012.0011.009.006.004.003.002.00.00
Fre
qu
en
cy
7
6
5
4
3
2
1
0
Statistics
jail sentence24
0
5.1250
5.44788
29.67935
Valid
Missing
N
Mean
Std. Deviation
Variance
Length of Sentence
![Page 34: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/34.jpg)
grade point average
4.24.03.83.63.43.23.02.8
jail
sen
ten
ce
20
10
0
-10
Scatterplot: Sentence on G.P.A.
![Page 35: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/35.jpg)
Regression Coefficients
Coefficientsa
17.853 12.097 1.476 .155
-3.534 3.368 -.223 -1.049 .306
(Constant)
grade point average
Model1
B Std. Error
UnstandardizedCoefficients
Beta
StandardizedCoefficients
t Sig.
Dependent Variable: jail sentencea.
Sentence = -3.5 G.P.A. + 18
![Page 36: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/36.jpg)
grade point average
4.24.03.83.63.43.23.02.8
jail
sen
ten
ce20
10
0
-10
Sent = -3.5 GPA + 18
“Least Squares” Regression Line
![Page 37: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/37.jpg)
Correlation: Sentence & G.P.A.
Correlations
1 -.223
. .306
23 23
-.223 1
.306 .
23 24
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
grade point average
jail sentence
grade pointaverage jail sentence
![Page 38: Regression & Correlation. Review: Types of Variables & Steps in Analysis.](https://reader035.fdocuments.us/reader035/viewer/2022062722/56649f2c5503460f94c47d7d/html5/thumbnails/38.jpg)
Interpreting Correlations
• r = -22 p = .31
• r2 = .05
G.P.A. “explains” 5% of the variance in length of sentence