Omnibus Tests in Multiple Regression Omnibus test

1 omnibus tests in multiple regression

1.1 model assumptions in multiple linear regression
1.2 omnibus f test regarding hypotheses on coefficients
1.3 example 1- omnibus f test on spss

1.3.1 anova
1.3.2 model summary
1.3.3 coefficients


1.4 example 2- multiple linear regression omnibus f test on r

1.4.1 coefficients

omnibus tests in multiple regression

in multiple regression omnibus test anova f test on coefficients, equivalent multiple correlations r square f test. omnibus f test overall test examines model fit, rejecting null hypothesis implies suggested linear model not significally suitable data. in other words, none of independent variables has explored significant in explaining dependant variable variation. these hypotheses examine model fit of common model: yi=β0 + β1 xi1 + ... +βk xik + εij

estimated e(yi|xi1....xik)=β0+β1xi1+...+βkxik ,where e(yi|xi1....xik) dependant variable explanatory i-th observation, xij j-th independent (explanatory) variable, βj j-th coefficient of xij , indicates influence on dependant variable y upon partial correlation y. f statistics of omnibus test is:



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{\displaystyle f={\frac {{\displaystyle \sum _{i=1}^{n}\left({\widehat {y_{i}}}-{\bar {y}}\right)^{2}}/{k}}{{\displaystyle {\sum _{j=1}^{k}}{\sum _{i=1}^{n_{j}}}\left(y_{ij}-{\widehat {y_{i}}}\right)^{2}}/{(n-k-1)}}}}

whereas, ȳ overall sample mean yi, ŷi regression estimated mean specific set of k independent (explanatory) variables , n sample size.

the f statistic distributed f (k,n-k-1),(α) under assuming of null hypothesis , normality assumption.

model assumptions in multiple linear regression

random sampling.
normal or approximately normal distribution of errors eij.
the errors eij explanatory equals zero>, e(eij)=0.
equal variances of errors eij. s omnibus f test ( levene f test).
no multi-collinearity between explanatory/predictor variables meaning: cov(xi,xj)=0 i≠j, or j.

the omnibus f test regarding hypotheses on coefficients

h0: β1= β2=….= βk = 0

h1: @ least 1 pair βj≠ βj

the omnibus test examines whether there regression coefficients non-zero, except coefficient β0. β0 coefficient goes constant predictor , not of interest. null hypothesis thought false , rejected reasonable amount of data, in contrary anova important test anyway. when null hypothesis cannot rejected, means data worthless. model has constant regression function fits regression model, means no further analysis need done. in many statistical researches omnibus significant, although part or of independent variables has no significance influence on dependant variable. omnibus useful imply whether model fits or not, doesn t offers corrected recommended model can fitted data. omnibus test comes significant if @ least 1 of independent variables significant. means other variable may enter model, under model assumption of non-colinearity between independent variables, while omnibus test still show significance, is: suggested model fitted data. significance of omnibus f test (shown on anova table) followed model selection, part of related selection of significant independent variable contribute dependant variable variation.

example 1- omnibus f test on spss

an insurance company intends predict average cost of claims (variable name claimamt ) 3 independent variables (predictors): number of claims (variable name nclaims ), policyholder age (variable name holderage), vehicle age (variable name vehicleage). linear regression procedure has been run on data, follows: omnibus f test in anova table implies model involved these 3 predictors can fit predicting average cost of claims , since null hypothesis rejected (p-value=0.000 < 0.01, α=0.01). rejection of omnibus test implies coefficients of predictors in model have found non-zero. multiple- r-square reported on model summary table 0.362, means 3 predictors can explain 36.2% average cost of claims variation.

anova

a. predictors: (constant), nclaims number of claims, holderage policyholder age, vehicleage vehicle age

b. dependent variable: claimamt average cost of claims

model summary

a. predictors: (constant), nclaims number of claims, holderage policyholder age, vehicleage vehicle age

however, predictors: vehicle age , number of claims has statistical influence , prediction on average cost of claims shown on following coefficients table , whereas policyholder age not significant predictor (p-value=0.116>0.05). means model without predictor may suitable.

coefficients

a. dependent variable: claimamt average cost of claims

example 2- multiple linear regression omnibus f test on r

the following r output illustrates linear regression , model fit of 2 predictors: x1 , x2. last line describes omnibus f test model fit. interpretation null hypothesis rejected (p = 0.02692<0.05, α=0.05). either β1 or β2 appears non-zero (or perhaps both). note conclusion coefficients: table β1 significant (p-value shown on pr(>|t|) column 4.37e-05 << 0.001). 1 step test, omnibus f test model fitting not sufficient determine model fit predictors.

coefficients

  estimate   std. error   t value   pr(>|t|)

(intercept)   -0.7451   .7319  .-1.018  0.343

x1   0.6186   0.7500   0.825   4.37e-05 ***

x2   0.0126   0.1373   0.092   0.929

residual standard error: 1.157 on 7 degrees of freedom

multiple r-squared: 0.644, adjusted r-squared: 0.5423

f-statistic: 6.332 on 2 , 7 df, p-value: 0.02692