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Analysis of Dependent Variables: Correlation and Simple Regression Zacariah Labby, PhD, DABR

Asst. Prof. (CHS), Dept. of Human OncologyUniversity of Wisconsin – Madison

Conflicts of Interest

None to disclose

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Purpose•Review basic statistics and identify appropriate use of statistics related to analyzing simple relationships between two variables:•Correlation statistics• Linear regression and model fitting

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STATISTICS OF CORRELATION

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Correlation: Review of Terminology•Dependent vs. Independent Variables•Standard plot: X is Independent

Y is Dependent•Linear vs. Monotonic• Linear: increase in X leadsto proportional increase in Y•Monotonic: increase in Xleads to some increase in Y

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Correlation: Review of Terminology•Variable Type•Continuous•Example: Ionization chamber charge collected vs. Dose delivered

•Discrete•Example: Number of patients seen vs. Calendar year

•Ordinal•Example: Severity of normal tissue toxicity vs. Prescription Level

•Categorical•Example: RECIST response classification vs. Radiologist Observer

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Correlation: Metrics of Interest • Four big categories of data•Continuous

•Discrete

•Ordinal

•Categorical

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Correlation: Metrics of Interest • Four big categories of data•Continuous

•Discrete

•Ordinal

•Categorical

Three major correlation metrics

Pearson’s r

Spearman’s ⍴

Fleiss’ κ

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Correlation: Metrics of Interest • Four big categories of data•Continuous

•Discrete

•Ordinal

•Categorical

Three major correlation metrics

Pearson’s r

Spearman’s ⍴

Fleiss’ κ

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Correlation: Pearson’s r• “Linear” or “Product-Moment” correlation•Applies only to continuous data•Parametric correlation• Tendency of dependent variable to increase linearly with the independent variable

•Key Point:• There is an assumed form to the relationship•Linear, and therefore also monotonic

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Correlation: Pearson’s r

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r = 1.00 r = 0.97 r = 0.76

r = 0.96 r = 0.96 r = 0.75

Correlation: Pearson’s r

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r = 1.00 r = 0.97 r = 0.76

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Correlation: Spearman’s ⍴• “Rank” correlation•Applies to continuous, discrete, or ordinal data•Non-parametric correlation• Tendency of dependent variable to increase with the independent variable

•Key Point:• There is no assumed relationship, only monotonicity

•Math: Pearson’s r of rank-transformed data

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Correlation: Spearman’s ⍴

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Correlation: Spearman’s ⍴

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Raw: (0,0)Rank: (1,1)

(X,Y) pairs

Correlation: Spearman’s ⍴

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Raw: (0.05,0.0025)Rank: (2,2)

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Correlation: Spearman’s ⍴

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Raw: (1,1)Rank: (20,20)

Pearson’s r of rank-transformed data: 1.00

Raw: (0,0)Rank: (1,1)

Raw: (0.05,0.0025)Rank: (2,2)

Correlation: Spearman’s ⍴

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r = 1.00⍴ = 1.00

r = 0.97⍴ = 0.97

r = 0.76⍴ = 0.90

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Correlation: Spearman’s ⍴

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r = 1.00⍴ = 1.00

r = 0.97⍴ = 0.97

r = 0.76⍴ = 0.90

r = 0.96⍴ = 1.00

r = 0.96⍴ = 0.99

r = 0.75⍴ = 0.89

Correlation: Which Metric?

Continuous variables; “When one goes up, does the other (reliably) go down?”Aug 1, 2016 Labby - AAPM 2016 20

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Disease Volume

Lung Volume

Z.E. Labby et al, J Thorac Oncol 8, (2013)

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Correlation: Which Metric?

Continuous variables; “When one goes up, does the other (reliably) go down?”Aug 1, 2016 Labby - AAPM 2016 21

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Z.E. Labby et al, J Thorac Oncol 8, (2013) Answer:Spearman’s ⍴

Correlation: Fleiss’ κ•Categorical correlation•Applies only to categorical data•Categorical data could be inherently ordinal

•Non-parametric correlation•How well do independent categories sort dependent categories?

•Math: number of dependent-independent pairs in agreement over the number expected by chance alone.

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Correlation: Fleiss’ κ•Example:• 5 radiologists contour tumors in• 31 patients•Response classification from baseline to post-chemo CT scans•Progressive Disease•Stable Disease•Partial Response•Complete Response

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Obs. 1 Obs. 2 Obs. 3 Obs. 4 Obs. 5

Progression 6 11 7 11 14

Stable 17 10 19 15 9

Partial 7 10 5 4 8

Complete 1 0 0 1 0

κ = 0.64Landis and Koch, Biometrics, 33,159–174 (1977)

Correlation vs. Agreement•Quick tangent…

Important question:Do you already know that the two variables will be correlated?

Example: Tumor volumes as assessed by Physician vs. Algorithm

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Correlation vs. Agreement•Especially with implicit independent variables (i.e., the true value remains unknown), correlation isn’t as meaningful•Correlation is only the strength of a relationship between two variables•Agreement is the actual 1:1 accuracy

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Bland and Altman, “Statistical methods for assessing agreement between two methods of clinical measurement,” Lancet 327, 307 (1986).

Correlation vs. Agreement

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Average of Physician and Algorithm

Difference

(Physician –Algorithm)

Mean

Mean + 2SD

Mean - 2SD

Bland and Altman, “Statistical methods for assessing agreement between two methods of clinical measurement,” Lancet 327, 307 (1986).

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Correlation vs. Agreement•Absolute agreement vs. Relative agreement•Absolute: plot raw differences•Relative: plot log differences

•Get mean, SD of log-transformed data, then apply exponential to get relative agreement bounds

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Bland and Altman, “Statistical methods for assessing agreement between two methods of clinical measurement,” Lancet 327, 307 (1986).

ln𝑥𝑦 = ln 𝑥 − ln 𝑦

SIMPLE MODELING

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Correlation vs. Agreement vs. Modeling•Correlation: Strength of relationship•Agreement: Accuracy of 1:1 match•Modeling: Quantifying the relationship

•Rules of Modeling:1. Prefer model with n-1 parameters to n2. Prefer model with k-1 independent variables to k3. Prefer linear model to curved model

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Crawley, Statistics: An Introduction using R, Wiley (2005)

Simple Linear Regression•Linear regression is linear in the coefficients, not necessarily in the independent variable

•Linear:

•Not Linear:

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𝒚 = 𝛼 + 𝛽𝒙𝒚 = 𝛼 + 𝛽𝒙,

𝒚 = 𝛼 + 𝑒.𝒙

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Simple Linear Regression• I was going to put the math here, but…

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Simple Linear Regression•Sources of Variance in the data

•Your model:•Reality:

•Variance in y can be explained by•Variance in x•Residual uncertainty (called 𝝐 )

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𝑦0 = 𝛼 + 𝛽𝑥0 + 𝜖0𝒚 = 𝛼 + 𝛽𝒙

Random (residual) error from fit for each 𝑥0

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Simple Linear Regression•Sources of Variance in the data•Explained Sum of squared errors (ESS)•Residual Sum of squared errors (RSS)• Total sum of squared errors (TSS)

•Coefficient of Determination: ESS/TSS•Proportion of total variation in y explained by the model

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𝑓 𝑥0 − 𝑦3 , + 𝑦0 − 𝑓 𝑥0, = 𝑦0 − 𝑦3 ,

Simple Linear Regression•Coefficient of Determination: ESS/TSS• Proportion of total variation in y explained by the model

•Has another name…R-squared!

•Pearson’s correlation coefficient• 𝑟 = 𝑅,�

•Drive home: Correlation quantifies strength of relationship, not relationship itself

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𝑅, =𝐸𝑆𝑆𝑇𝑆𝑆

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Simple Linear Regression•Making predictions• From analysis, derive best-fit values of fit 𝛼:, 𝛽<, etc.•Predict new values according to 𝑦=>? = 𝛼: +𝛽<𝑥=>?

•However, models have uncertainty!•Variance estimates can be provided for 𝛼:, 𝛽<, etc. (e.g., 𝜎:A)

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Simple Linear Regression•Confidence Bands•Variance associated with mean predicted response

•Prediction Bands•Variance associated with single new prediction• Takes into account residual errors in linear model

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Var 𝛼: + 𝛽<𝑥=>?

Var 𝛼: + 𝛽<𝑥=>? + 𝜎:E,(in some ways, this is like the difference

between standard deviation and

standard error)

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Simple Linear Regression

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Data Points

Linear Model

95% Conf. Bands

95% Pred. Bands

Example•Task: Department administrator asks you to figure out the relationship between patient census and required RadTechhours.•Question 1: what kind of relationship would we expect?•Probably Linear with some residual uncertainty

•Question 2: which correlation metric would you use?•Pearson’s r

•Question 3: how would you quantify the relationship?•Simple linear regression

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Example•Regression model

•Predict tomorrow’s RadTech staffing level if you know the patient workload•Could staff at the upper 95% prediction band?

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StaffingRequirements = 𝛼: + 𝛽<PatientWorkload𝛼: = Fixed StaffOverhead𝛽< = ScalableCoefficient

A plug for “R”…•R is a free software package for data analysis and is very common in the statistics community.

•Good text for learning R and basic stats:•Statistics: An Introduction using R by Michael J. Crawley, published 2005 by John Wiley & Sons, Ltd

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A plug for “R”…model=lm(y~x)

conf.bands=predict(model,interval=‘conf’)

pred.bands=predict(model,interval=‘pred’)

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Use your Biostatisticians•Many large centers have at least one biostatistician on staff• In many centers, free consultations for•Experimental design•Simple clinical trials•Data analysis questions

•Prevent headaches and lost costs for rework and rejected papers

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