Inference in Regression Coefficients

8/19/2019 Inference in Regression Coefficients

http://slidepdf.com/reader/full/inference-in-regression-coefficients 1/6

Written b : Asst. Pro . Xandro Alexi A. Nieto o UST – Facult o Pharmac

INFERENCE IN REGRESSION COEFFICIENTS

- tests whether ! i ! 0; i = 1, 2, 3, …k

SIMPLE LINEAR REGRESSION- used to estimate the dependent variable Y for given set of independent variable X.

Y = a + bX + ! or

Y = ! 0 + ! 1X + ! ; where

!! !! !"

! ! !

! !!! !

!; !! !

!

!

! !!!

!

; and ! ! ! ! !

- inference in ! 1 may be performed to determine if it is significantly different from zero ( ! 1 ! 0 ), using

! !!!!!

!!!!!

!! !

!

; with df = n – 2

- a linear relationship (linearity) exists between Y and Xi if the -value of ! 1 (using t-test) < !.

- R 2 is the proportion of the total variance (s2 ) of Y that can be explained by the linear regression of Y on X.

Example:

Using the example about the file, HCTRBC.sav , find the linearregression model that estimates the RBC (Y, in x1012/L), given thehematocrit (X, in % vol) of a patient.

Find Y = ! 0 + ! 1X + !

!! !! !" ! ! !

! !!! !

! !

!! !!

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!

!

Estimate the RBC ! of a patient with hematocrit of 43.2 %.

Find the residual ! of the simple linear regression model if a patient has HCT of 40.7%.

ID

HCT(% vol)

X

RBC(x1012/L)

Y X2 Y 2 XY

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MULTIPLE LINEAR REGRESSION

Y = ! 0 + ! 1X1 + ! 2X2 + … + ! k Xk + ! or ! ! !! ! !! !!!

!!! ! !

- linear relationship (linearity) exists between Y and Xk if the p-value of the ! k < !, using the individual t-tests ofthe ANOVA result.

- Hypotheses are as follows:

Ho: !! ! !.

Ha: !! ! !. Diagnostic checking of the linear regression model may be applied by checking if:

• the residuals ! are normally distributed (Kolmogorov-Smirnov Test of Normality)

Ho: The residuals ! are normally distributed.

Ha: The residuals ! are not normally distributed.• the residuals have constant variance (by using Levene’s test or Bartlett’s test)

Ho: The variances are equal.Ha: The variances are not equal.

Examples:1. A researcher wants to determine if which among the variables (mother and father’s height; taller grandfather’s height)

determine a son’s height (expressed in inches). The data is in heights.sav . Test all hypotheses at ! = 0.05.

.

Summary of the Findings:

_________________________________________________________

_________________________________________________________

_________________________________________________________

_________________________________________________________

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2. (bloodlead.sav) A group of researchers wanted to determine the factors that contributes to the amount of blood lead leve(in "g/dL) in radiator repair workers. Data such as number of radiators repaired per day, years of employment, and renafunction tests [FBS (in mmol/L), creatinine (in "mol/L), crea (in mg/dL), BUN (in mmol/L), presence of protein in urineand eGFR (in mL/min/1.73m)] were gathered. Conduct a multiple regression model to determine the factors thacontribute to the amount of blood lead level in radiator repair workers. Use 5% level of significance.

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Multiple linear regression R 2 = ___________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

Variables Coefficient t stat p-value

Number of Radiators repaired per day

Years of employmentRenal Function Tests

FBS (mmol/L)

Creatinine (µmol/L)

Crea (mg/dL)

BUN (mmol/L)

eGFR (ml/min/1.72m)




MULTIPLE LOGISTIC REGRESSION

!" !

!!!! !! ! !! !! ! !! !! !!!! !! , or

!" !

!!!! !! ! !! !!

!

!!! ! ! where p = P(Y=1)

Consider that!

!!!! !

!!!!!!!!!!!!!!!!!!

- used when the dependent variable Y is dichotomous variable, when at least one of the independent variables X

,

i !1,2,…,k , is interval/ratio.

- validity of the model may be tested using the Hosmer and Lemeshow test, in which:Ho: the data fits the model.

Ha: The data does not fit the model.

Example 1: An oncologist is interested to determine the variables that lead to papillary tumor growth, cancerous cells whichare found in the throat. Data from 40 patients who may have lived with exposure to radioactive iodine in the last 5 years and

who have had thyroiditis in the last six months is at thyroiditis.sav.

Model Fit Test:

Ho: ________________________

Ha: ________________________

Test Statistic: __________

p-value: ______________

Conclusion: __________________

Which of the variables significantly

coefficients significantly differ from

zero?

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The data is fit for logistic regression !!

!! !!!"#!! ! !!!"! .

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

Example 2: (renalcast.sav) A group of researches wanted to determine the variables that leads to renal cast formation oconstruction workers. Years in the occupation, if painting is included in the occupation, and urinary findings, such as BUNuric acid, PH, and presence of bacteria were recorded. Conduct a multiple logistic regression model to determine the

variables that leads to renal cast formation of construction workers. Use 5% level of significance.

Model Fit Test:

Ho: ________________________ Ha: ________________________

Test Statistic: __________ p-value: ______________

Conclusion: __________________

Summarize your findings using the table below:

The data is fit for logistic regression !!

!! !!!!!!!!!!!!!!!!!!! ! !!!!!!!!!!!!!!!!!! .

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

__________________________________________________________________________________________

Variables Coefficient " stat p-value Odds Ratioestimate

Nuclear Location

Gender

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Dental or chest xray in the last 2 years

High dosage of xray in the last 2 years

Family history

Variables Coefficient " 2 stat p-value Odds Ratio

estimate

Years in Occupation

Painting

EFG

Uric Acid

pH

Bacteria

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Inference in Regression Coefficients