1. Correlation and Regression Mathematics & Statistics Help University of Sheffield

2. Learning outcomes By the end of this session you should know about:
Approaches to analysis for simple continuous bivariate data
By the end of this session you should be able to:
Construct and interpret scatterplots in SPSS
Identify when it is appropriate to use correlation
Calculate a correlation coefficient in SPSS
Interpret a correlation coefficient
Identify when it appropriate to use linear regression
Run a simple regression model in SPSS
Interpret the results of a linear regression model

3. Download the slides from the MASH website
MASH > Resources > Statistics Resources > Workshop materials

4. Association between two continuous variables: correlation or regression?
Two basic questions:
Is there a relationship?
No causation is implied, simply association
Use CORRELATION
How can we use the value of one variable to predict the value of the other variable?
May be causal, may not be
Use REGRESSION

5. Correlation: are two continuous variables associated?
When examining the relationship between two continuous variables ALWAYS look at the scatterplot, to see visually the pattern of the relationship between them

6. Scatterplot: Relationship between two continuous variables:
Outlier
Linear
Explores the way the two co-vary (correlate):
Positive / negative
Linear / non-linear
Strong / weak
Presence of outliers
The scatter graph or scatterplot is one of the most common graphs in statistics. It is used to explore the relationship between two continuous variables. Is there a linear or non-linear pattern? The summary statistic we calculate to represent this relationship is called the Pearson’s correlation co-efficient. Is the correlation positive or negative? Is the correlation, strong or weak? This graph will help us answer these questions. It will also help us detect any outliers, that is observations that are outside the norm, outside the pattern that the other observations show.
As usual, the scatter graph can be built using SPSS or Excel or any other statistical software.

7. Scatterplots

8. Correlation Coefficient r
Measures strength of a linear relationship between 2 continuous variables, can take values between -1 to +1
r = 0.9
r = 0.01
r = -0.9

9. Correlation: Interpretation
An interpretation of the size of the coefficient has been described by Cohen (1992) as:
Cohen, L. (1992). Power Primer. Psychological Bulletin, 112(1)
Correlation coefficient value
Effect size
-0.3 to +0.3
Weak
-0.5 to -0.3 or 0.3 to 0.5
Moderate
-0.9 to -0.5 or 0.5 to 0.9
Strong
-1.0 to -0.9 or 0.9 to 1.0
Very strong

10. Relationship is not assumed to be a causal one – it may be caused by other factors
Does chocolate make you clever or crazy?
A paper in the New England Journal of Medicine claimed there was a relationship between chocolate and Nobel Prize winners

11. Chocolate and serial killers
What else is related to chocolate consumption?

12. Dataset for today: Birthweight_reduced_data
Factors affecting birth weight of babies
Mother smokes = 1
Standard gestation = 40 weeks

13. Exercise 1: Gestational age and birthweight
Draw a line of best fit through the data (with roughly half the points above and half below). Describe the relationship
Is the relationship:
strong/ weak?
positive/ negative?
linear?

14. Exercise 2: Interpretation
Interpret the following correlation coefficients using Cohen’s classification and explain what they mean. Which correlations seem meaningful?
Relationship
Correlation
Average IQ and chocolate consumption
0.27
Road fatalities and Nobel winners
0.55
Gross Domestic Product and Nobel winners
0.70
Mean temperature and Nobel winners
-0.60

15. Scatterplot in SPSS
Graphs  Legacy Dialogs  Scatter/Dot

16. Scatterplot in SPSS
Graphs  Legacy Dialogs  Scatter/Dot

17. Use Spearman’s correlation for ordinal variables or skewed scale data
Correlation in SPSS
Analyze  Correlate  Bivariate  Pearson
Use Spearman’s correlation for ordinal variables or skewed scale data

18. Scatterplot and correlation
SPSS output using reduced baby weight data set
Pearson correlation
r = 0.708
Strong relationship

19. Hypothesis test for the correlation coefficient
Can be done, the null hypothesis is that the population correlation r = 0
Not recommended as it is influenced by the number of observations
Better to use Cohen’s interpretation

20. Hypothesis test: Influence of sample size
Value at which the correlation coefficient becomes significant at the 5% level (i.e. p<0.05) 10 20 50
100
150
0.63
0.44
0.28
0.20
0.16

21. And so what do correlations of 0.63 (n=10) and 0.16 (n=150) look like?
Correlation=0.63, p=0.048 (n=10)
Correlation=0.16, p=0.04 (n=150)

22. Points to note Do not assume causality
Be careful comparing the correlation coefficient, r, from different studies with different n
Do not assume the scatterplot looks the same outside the range of the axes
Use Cohen’s scale to interpret, rather than the p- value
Always examine the scatterplot!

24. Exercise 3a: Scatterplot
Use Recode > Transform into Different Variables to construct a variable for maternal smoking status (non-smoker / smoker)
Construct a scatterplot for birthweight and gestational age? Use Set Markers by to distinguish between smokers and non-smokers
Is there evidence of a linear relationship
Interpret the correlation coefficient. What does it mean?
Note:
Think about which variable should be on the x axis (horizontal) and which should be on the y axis( vertical)
If you double-click on the graph you can open the Graph dialog window and edit the chart, for example change the colours used for smokers and non-smokers

25. Exercise 3b: Scatterplot & Correlation
Construct a scatterplot and calculate Pearson’s correlation coefficient for birthweight and maternal pre- pregnancy weight?
Is there evidence of a linear relationship
Interpret the correlation coefficient. What does it mean?
Note: think about which variable should be on the x axis (horizontal) and which should be on the y axis( vertical)

26. Association between two continuous variables: correlation or regression?
Two basic questions:
Is there a relationship?
No causation is implied, simply association
Use CORRELATION
How can we use the value of one variable to predict the value of the other variable?
May be causal, may not be
Use REGRESSION

27. Simple linear regression
Regression quantifies the relationship between two continuous variables
It involves estimating the best straight line with which to summarise the association
The relationship is represented by an equation, the regression equation
It is useful when we want to
look for significant relationships between two variables
predict the value of one variable for a given value of the other

28. Independent / dependent variables
Does attendance have an association with exam score?
Does temperature have an impact on the growth rate of a cell culture?
DEPENDENT (outcome) variable (y)
INDEPENDENT
(explanatory/
predictor) variable (x)
affects
You will need to distinguish between independent (explanatory) variables and dependent (outcome) variables regarding the research question. The explanatory variables are thought to have an effect on the dependent variable and the distinction between the two is important when carrying out statistical analysis. For example, if you were investigating the relationship between attendance is exam score, the independent variable is the attendance and the dependent variable is the exam score.

29. Does gestational age have an association with birth weight?

30. Regression
y = a + b x
Simple linear regression looks at the relationship between two continuous variables by producing an equation for a straight line of the form
You can use this to predict the value of the dependent (outcome) variable for any value of the independent (explanatory) variable
Independent variable
Dependent variable
Intercept
Slope

31. Birth weight example equation
Birth weight (y) = * gestational age (x)
here, a = (intercept)
b = (slope)
i.e. for every extra week of gestation, birth weight increases by 0.16 kgs

32. b = 0.16 so extra 0.16 kgs for every extra week of gestation
Birth weight example - Slope
Slope b is the average change in the Y variable for a change of one unit in the X variable
b = 0.16 so extra 0.16 kgs for every extra week of gestation

33. Birth weight example - Intercept
Y Response variable (dependent variable)
X Predictor / explanatory variable (independent variable)

34. Estimating the best fitting line
We try to fit the “best” straight line
The standard way to do this is using a method called least squares using a computer
Residuals = differences between observed and predicted values for each observation
Least squares method chooses a line so that the sum of the squared residuals (averaged over all points) is minimised

35. Line of best fit
Residuals =
observed - predicted

36. Hypothesis testing in regression
Regression finds the best straight line with which to summarise an association
It is useful when we want to look for significant relationships between variables
The slope is tested for significance. If there is no relationship, the gradient of the line (b) would be 0; i.e. the regression line would be a horizontal line crossing the y axis at the average value for the y variable

37. Regression in SPSS
Analyse  Regression  Linear

38. Output from SPSS: key regression table
P – value < 0.001
Y = X
As p < 0.05, gestational age is a significant predictor of birth weight
Weight increases by 0.16 kgs for each week of gestation

39. Output from SPSS: ANOVA table
ANOVA compares the null model (mean birth weight for all babies) with the regression model
Null model: y = 3.31
Regression model: y = x

40. Output from SPSS: ANOVA table
Does a model containing gestational age predict significantly more accurately than just using the mean birth weight for all babies?
Yes as p < 0.001
Total: number of subjects included in the analysis – 1

41. How reliable are predictions? Using R2
How much of the variation in birth weight is explained by the model including Gestational age?
Proportion of the variation in birth weight explained by the model R2 = = 50%
Predictions using the model are fairly reliable.
Which other variables may help improve the fit of the model?
Compare models using Adjusted R2, as this adjusts for the number of variables in the model

42. Exercise 4
Investigate whether mother’s pre-pregnancy weight and birth weight are associated using a simple linear regression

43. Exercise 4: regression Adjusted R2 =
Does the model result in reliable predictions?
ANOVA p-value =
Is the model an improvement on the null model (where every baby is predicted to be the mean weight)?

44. Exercise 4: Regression Pre-pregnancy weight coefficient and p-value:
Regression equation:
Interpretation:

45. Assumptions for regression
Plot to check
The relationship between the independent and dependent variable is linear
Original scatter plot of the independent and dependent variable
Homoscedasticity: The variance of the residuals about predicted responses should be the same for all predicted responses.
Scatterplot of standardised predicted values and residuals. There should be no obvious patterns
The residuals are normally distributed
Plot the residuals in a histogram

46. Checking assumptions: normality of residuals
observed value minus value predicted by the model (fitted value)
Yobs - Yfit
i.e. the vertical lines on the plot below
It is the residuals that need to be normally distributed, not the data

47. Checking assumptions: normality of residuals
Use standardised residuals to check the assumptions. Outliers are those values < -3 or > 3
Select histogram of residuals
Scatterplot of predicted vs residuals

48. Checking assumptions: normality
Histogram looks approximately normally distributed When writing up, just say ‘normality checks were carried out on the residuals and the assumption of normality was met’

49. Predicted values against residuals
Are there any patterns as the predicted values increases?
There is a problem with Homoscedasticity if the scatter is not random. These shapes are bad:

50. Exercise 5
Re-run the regression model, but this time, produce the residual plots. Do you think that the assumptions of normality of residuals and homogeneity of variance are met?

51. What if assumptions are not met?
Regression is fairly robust to violations of the assumptions
If the residuals are heavily skewed or the residuals show different variances as predicted values increase, the data needs to be transformed
Try taking the natural log (ln) of the dependent variable first. Then repeat the analysis and check the assumptions

52. Caveats
Do not use the graph or regression model to predict outside of the range of observations
Do not assume just because you have an equation that means that X causes Y
As with correlation, it is always a good idea to have a look at the scatterplot

53. Effect of outliers on regression model
Original regression model:
y = x
R2 =0.502
Adjusted regression model:
y = x
R2 =0.323

54. Multiple regression
Multiple regression has several categorical or scale independent variables:
𝑦=𝛼+ 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 +…+ 𝛽 𝑖 𝑥 𝑖
Effect of other variables is removed (controlled for) when assessing relationships
You can only include binary categorical using:
Analyse  Regression  Linear
If you want to adjust for a categorical variable with more than 2 levels you need to use the General Linear Model procedure
Analyse  General Linear Model  Univariate
(not covered here)

55. Multiple regression
Example: What factors affect the birth weight of babies?
Dependent: Birth weight
Possible independents: Gestational age, mothers’ pre-pregnancy weight, mothers height, mother smoking etc

56. Relationships with categorical
Identify smokers by different markers on the scatterplot
Graphs  Legacy Dialogs  Scatter/Dot

57. Adding smoking status
Identify smokers by different markers on the scatterplot
Is there a difference between smokers and non-smokers?

58. Regression output from SPSS
Y = (gestation) – 0.298(smoker)
Both
p-values < 0.05
R2 has increased from to so 56.3% of variation explained with gestation and smoking.
Gestational age (p < 0.001) and smoking status (p=0.024) are significant predictors of birth weight. Weight increases by 0.16 kgs for each week of gestation and decreases by 0.30 kgs for smokers
Note: Smoker = 1 and Non-smoker = 0

59. Effect of smoking status
Binary variables affect the intercept only
Note that if you have an interactions between a binary and scale variable the lines for the two groups will not be parallel

60. Comparing models using R2
Adding predictor variables will always increase R2. It’s the size of the change that’s important
Use adjusted R2 as it makes an adjustment for the number of variables in a model

61. Multiple regression
In addition to the standard linear regression checks, relationships BETWEEN independent variables should be assessed
Multicollinearity is a problem where continuous independent variables are too related (r > 0.8)
Relationships can be assessed using scatterplots and correlation for continuous variables

62. Exercise 6: correlations
Produce a correlation matrix for the correlations between Birthweight, Gestational age, Maternal height and Maternal pre-pregnancy weight: Analyse > Correlate > Bivariate & add the 4 variables to the Variables box:

63. Model selection
If models are to be used for prediction, only significant predictors should be included unless they are being used as controls
Methods include forward, backward and stepwise regression
Backward means that the predictor with the highest p-value is removed and the model re-run. Keep going until only significant predictors are left

64. Important point
By default, SPSS only includes cases with no missing values on any variable included in multiple regression. This can seriously reduce the number of cases being used
To change this, select ‘Options’ from the linear regression options and then ‘Exclude cases pairwise’

65. Exercise 7
With birthweight as the outcome, run a series of regression models:
Model 1: Gestational age
Model 2: Gestational age and maternal smoking status
Check the assumptions and interpret the output of
Does the model give more reliable predictions than the model with just gestational age?
Model 3: gestational age, maternal smoking status, maternal pre-pregnancy weight
Model 4: gestational age, maternal smoking status, maternal pre-pregnancy weight, maternal height
Note you will need to create a variable for smoking status based on the number of cigarettes that the mother smokes (assuming that 0 cigarettes indicates someone who does not smoke)

66. Exercise 7: model 1 summary
Variable
Coefficient (β)
P-value
Significant?
Constant
Gestation
Adjusted R2 =
Interpretation:

67. Exercise 7: model 2 summary
Variable
Coefficient (β)
P-value
Significant?
Constant
Gestation
Smoker
Adjusted R2 =
Interpretation:

68. Exercise 7: model 3 summary
Variable
Coefficient (β)
P-value
Significant?
Constant
Gestation
Smoker
Pre-pregnancy weight
Adjusted R2 =
Interpretation:

69. Exercise 7: model 4 summary
Variable
Coefficient (β)
P-value
Significant?
Constant
Gestation
Smoker
Pre-pregnancy weight
Height
Adjusted R2 =
Interpretation:

70. Exercise 7: Compare p-values
Model
Gestation
Smoking
Weight
Height
Model 1:
P < 0.001
Model 2:
Model 1 + Smoker
0.028
Model 3:
Model 2 + Weight
Model 4:
Model 3 + Height

71. Exercise 7: Compare R2 Model R2 Adjusted R2 Model 1: Gestation 0.499
0.486
Model 2:
Model 1 + Smoker
0.558
0.535
Model 3:
Model 2 + Weight
Model 4:
Model 3 + Height

72. Regression summary
Use correlation to look at relationships between dependent and independent variables
Scatterplots to look for a linear relationships
Use regression to quantify the relationship between variables
Check normality of residuals
Check scatterplot of predicted vs residuals
Interpret significance, coefficients and R2

73. Learning outcomes You should now know about: You should be able to:
Approaches to analysis for simple continuous bivariate data
You should be able to:
Construct and interpret scatterplots in SPSS
Identify when it is appropriate to use correlation
Calculate a correlation coefficient in SPSS
Interpret a correlation coefficient
Identify when it appropriate to use linear regression
Run a simple regression model in SPSS
Interpret the results of a linear regression model

74. Maths And Statistics Help
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75. Resources: All resources are available in paper form at MASH or on the MASH website

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