Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study...
-
Upload
zoe-kelley -
Category
Documents
-
view
221 -
download
6
Transcript of Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study...
![Page 1: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/1.jpg)
Exploring Relationships:
Correlations & Multiple Linear Regression
Dr James Betts
Developing Study Skills and Research Methods (HL20107)
![Page 2: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/2.jpg)
Lecture Outline:
•Correlation Coefficients
•Coefficients of Determinations
•Prediction & Regression
•Multiple Linear Regression
•Assessment Details.
![Page 3: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/3.jpg)
Statistics
Descriptive Inferential
Correlational
Relationships
GeneralisingOrganising, summarising & describing data
Significance
![Page 4: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/4.jpg)
Correlation• A measure of the relationship (correlation) between
interval/ratio LOM variables taken from the same set of subjects
• A ratio which indicates the amount of concomitant variation between two sets of scores
• This ratio is expressed as a correlation coefficient (r):
Perfect Negative
Relationship
Perfect Positive
RelationshipNo
Relationship+_
Strong Moderate Weak StrongModerateWeak
-1 +10 +0.7+0.3+0.1-0.7 -0.3 -0.1
![Page 5: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/5.jpg)
Correlation Coefficient & ScatterplotsDirection
Variable X (e.g. VO2max).
Var
iabl
e Y
(e.g
. 10
km r
un ti
me)
Variable X (e.g. VO2max)
Var
iabl
e Y
(e.g
. Exe
rcis
e C
apac
ity)
.
![Page 6: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/6.jpg)
Correlation Coefficient & Scatterplots
Variable X (e.g. VO2max)
Var
iabl
e Y
(e.g
. Exe
rcis
e C
apac
ity)
.Variable X (e.g. Age)
Var
iabl
e Y
(e.g
. Str
engt
h)
Form
![Page 7: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/7.jpg)
Correlation Coefficient & Scatterplots
Variable X (e.g. VO2max)
Var
iabl
e Y
(e.g
. Exe
rcis
e C
apac
ity)
.
Significance
Variable X (e.g. VO2max)
Var
iabl
e Y
(e.g
. 100
m S
prin
t tim
e)
.
![Page 8: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/8.jpg)
Correlation Coefficient & Scatterplots
Variable X (e.g. VO2max)
Var
iabl
e Y
(e.g
. Exe
rcis
e C
apac
ity)
.Variable X (e.g. VO2max)
Var
iabl
e Y
(e.g
. 100
m s
prin
t tim
e)
.
Significance
![Page 9: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/9.jpg)
Methods of Calculating r• Any method of calculating r requires:
– Homoscedacity (i.e. equal scattering)– Linear data (curvilinear data requires eta η)
• Parametric data (i.e. raw data >ordinal LOM and either
normal distribution or large sample) permits the use of ‘Pearson’s Product-Moment Correlation’
• If raw data violates these assumptions then use ‘Spearman’s Rank Order Correlation’ instead.
![Page 10: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/10.jpg)
X = Alcohol Units Y = Skill Score X2 Y2 XY
15 4 225 16 60
14 6 196 36 84
10 4 100 16 40
9 8 81 64 72
8 7 64 49 56
8 8 64 64 64
7 10 49 100 70
6 9 36 81 54
4 14 16 196 56
2 12 4 144 24Totals=
Pearson’s Product-Moment Correlation
![Page 11: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/11.jpg)
r = nXY-(X)(Y)
[nX2-(X)2] [nY2-(Y)2
Pearson’s Product-Moment Correlation
![Page 12: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/12.jpg)
X = Alcohol Units Y = Skill Score Rank X Rank Y D D2
15 4 10 1.5 8.5 72
14 6 9 3 6 36
10 4 8 1.5 6.5 42
9 8 7 5.5 1.5 2.3
8 7 5.5 4 1.5 2.3
8 8 5.5 5.5 0 0
7 10 4 8 4 16
6 9 3 7 4 16
4 14 2 10 8 64
2 12 1 9 8 64Total=
Spearman’s Rank-Order Correlation
![Page 13: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/13.jpg)
Spearman’s Rank-Order Correlation
r = 1 - 6D2
n(n2-1)
![Page 14: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/14.jpg)
![Page 15: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/15.jpg)
Correlations
1 -.860**
.001
10 10
-.860** 1
.001
10 10
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
VAR00001
VAR00002
VAR00001 VAR00002
Correlation is significant at the 0.01 level (2-tailed).**. Correlations
1.000 -.927**
. .000
10 10
-.927** 1.000
.000 .
10 10
Correlation Coefficient
Sig. (2-tailed)
N
Correlation Coefficient
Sig. (2-tailed)
N
VAR00001
VAR00002
Spearman's rhoVAR00001 VAR00002
Correlation is significant at the 0.01 level (2-tailed).**.
SPSS Correlation Outputs
![Page 16: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/16.jpg)
Coefficient of Determination (r2 x 100)• AKA ‘variance explained’, this figure denotes how much
of the variance in Y can be explained/predicted by X
e.g. to predict long jump distance (Y) from maximum sprint speed (X)
r = 0.8
r2 = 64%
Y X
![Page 17: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/17.jpg)
Correlation versus Regression
• By attempting to predict one variable using another, we are now moving away from simple correlation and moving into the concept of regression
Correlation =
Regression =
![Page 18: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/18.jpg)
Linear Regression • The equation for a linear relationship can be expressed as:
Y= a + bX -where: a = the y intercept; and b = the
gradient
Variable X (e.g. VO2max)
Var
iabl
e Y
(e.
g. E
xerc
ise
Cap
acit
y)
.
![Page 19: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/19.jpg)
![Page 20: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/20.jpg)
![Page 21: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/21.jpg)
SPSS Regression OutputLinear Regression
5.00 10.00 15.00
AlcoholUnits
5.00
7.50
10.00
12.50
Ski
llSco
re
SkillScore = 13.92 + -0.69 * AlcoholUnitsR-Square = 0.74
![Page 22: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/22.jpg)
Extrapolation versus Interpolation
Variable X (e.g. VO2max)
Var
iabl
e Y
(e.
g. E
xerc
ise
Cap
acit
y)
.
Remember that the accuracy of your equation depends upon the
linear relationship you observed ?
Interpolation =
Extrapolation =
![Page 23: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/23.jpg)
Multiple Linear Regression • We saw earlier how maximum sprint speed (X) can
predict/explain 64% of variance in long jump distance (Y) Y X
r2 = 64%
…but can Y be predicted any more effectively using more than one independent variable (i.e. X1, X2 , X3, etc)?
![Page 24: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/24.jpg)
Multiple Linear Regression • However, we can often predict Y effectively just using a
specific subset of X variables (i.e. a reduced model) Y X1
X2 Event Experience
![Page 25: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/25.jpg)
Multiple Linear Regression • ‘Best Subset Selection Methods’ involve calculation of r for every possible combination of IVs• Stepwise regression methods involve gradually either adding or removing variables and monitoring the impact of each action on r.
– Standard methods add and remove variables– Forward selection methods begin with 1 IV and add more– Backwards elimination methods begin with all IVs and remove
• The order in which IVs are added/removed is critical as the variance explained solely by any one will be entirely dependent upon the presence of others.
![Page 26: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/26.jpg)
![Page 27: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/27.jpg)
Model Summary
.860a .740 .708 1.74391Model1
R R SquareAdjustedR Square
Std. Error ofthe Estimate
Predictors: (Constant), AlcoholUnitsa.
Excluded Variablesb
.072a .374 .720 .140 .994
.208a 1.150 .288 .399 .950
BodyMass
Age
Model1
Beta In t Sig.Partial
Correlation Tolerance
CollinearityStatistics
Predictors in the Model: (Constant), AlcoholUnitsa.
Dependent Variable: SkillScoreb.
SPSS Multiple Linear Regression Output
![Page 28: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/28.jpg)
Summary: Exploring Relationships•The relationship between two variables can be expressed as a correlation coefficient (r)
•The coefficient of determination (r2) denotes the % of one variable that is explained by another
•Linear regression can provide an equation with which to predict one variable from another
•Multiple linear regression can potentially improve this prediction using multiple predictor variables.
![Page 29: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/29.jpg)
Coursework Project (40 % overall grade)• Your coursework will require you to address
2 out of 3 research scenarios that are available on the unit webpage
• For each of the 2 scenarios you will need to:
– Perform a literature search in order to provide a
comprehensive introduction to the research area
– Identify the variables of interest and evaluate the
research design which was adopted
– Formulate and state appropriate hypotheses…
![Page 30: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/30.jpg)
• Cont’d…– Summarise descriptive statistics in an appropriate
and well presented manner– Select the most appropriate statistical test with
justification for your decision– Transfer the output of your inferential statistics
into your word document– Interpret your results and discuss the validity and
reliability of the study– Draw a meaningful conclusion (state whether
hypotheses are accepted or rejected).
![Page 31: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/31.jpg)
Coursework Details (see unit outline)• 2000 words maximum (i.e. 1000 for each)
• Any supporting SPSS data/outputs to be appended
• To be submitted on Thursday 6th May
Assessment Weighting
Evaluation & Analysis (30 %)
Reading & Research (20 %)
Communication & Presentation (20 %)
Knowledge (30 %)
![Page 32: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/32.jpg)
Coursework Details• All information relating to your coursework (including
the relevant data files) are accessible via the module web page:
http://people.bath.ac.uk/jb335/Y2%20Research%20Skills%20(FH200107).html
Web address also referenced on shared area
Electronic copy to be included with submission.
Any further questions/problems can be raised in the CW revision lecture/labs after Easter
![Page 33: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/33.jpg)
Timed Practical Computing Exercise (20 % overall grade)
• This test will involve analysis/interpretation of the resultant data assessed via short answer questions
• Practice session Wednesday 14th April
• Duration = 80 min (2 groups)
• I will Email specific details after Easter.
![Page 34: Exploring Relationships: Correlations & Multiple Linear Regression Dr James Betts Developing Study Skills and Research Methods (HL20107)](https://reader035.fdocuments.in/reader035/viewer/2022062511/5515e124550346d46f8b4d03/html5/thumbnails/34.jpg)
bothIVs
unpaired
BothIVs
paired
>2 variables
2 variables
>2 groups
2 groups
>2 observations
2 observations
>1 observed frequency
1observedfrequency
Looking for differences between categories/frequencies?
(i.e. nominal LOM)
Goodnessof Fit χ2
Looking for differences within the same group
of subjects? (i.e. paired data)
Looking for differences between 2 separate groups of subjects? (i.e. unpaired data)
Looking for relationships?Looking for differences
with >1 independent variable?
Contingency χ2
Pairedt-test
1-way paired
ANOVA
Independent t-test
1-wayunpairedANOVA
Pearson’sr
Multiple Linear
Regression
2-waypaired
ANOVA
2-wayunpairedANOVA
1 IV paired1 IV
unpaired
2-waymixed model
ANOVA
Wilcoxon test
Friedman’stest
Mann-Whitneytest
KruskalWallis
test
Spearman’sr
Post-Hoc Tests
non-parametric
Start Here
If multiple DVs are involved then use MANOVA