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Comparison Of Anova And Linear Regression

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Question:

Analysis of Variance test is designed to examine differences among comparison groups or levels of a variable. Next, students will begin exploring relationships and associations using correlation and regression. To prepare for this material, do a little research online and discuss what the difference is between ANOVA and Linear Regression. A “bird in my window” once told me that there was no difference between these two tests. Was this bird lying?
 
 

Answer:

Concepts of Linear regression and analysis of variance or ANOVA are statistical techniques of understanding the behaviour of one variable with respect to another one. Linear regression relates the variation of a dependent variable to that of one or more independent variables. Analysis of variance is employed to determine whether there is any significant difference between two or more levels of one or more factors.

Consider a variable “Y” dependent on the variables “X1” and “X2”. Linear regression attempts to estimate the functional relationship of “Y” with “X1” and “X2”. The basic assumption of linear regression includes that the dependent or response variable follows a normal distribution and the errors in predicted values of “Y” from the actual values all identically and independently follow standard normal distribution. The method then estimates the regression coefficients, “β1” and “β2” for the independent variables which explain the changes in values of “Y” due to unit change in the corresponding independent variable. Linear regression is therefore a tool to estimate the relationship as well as predict the response variable on the basis of one or more independent or predictor variables (Lowry, 2014). Suppose it is of interest to explain and predict the lung function of a smoker as quantified by the lung capacity, by the number of packets of cigarettes smoked per year and the number of years of being a smoker. Then a linear equation on the predictor average packets smoked per year and number of years being a smoker is fitted on the basis of available data points. The fitted regression equation then depending upon its “goodness of fit” can predict lung capacity on for given predictor values.

 


ANOVA according to Gliner et al. (2017) is, “A statistical test conducted when there is a nominal independent variable with three or more levels and a scale/continuous dependent variable.” The tool can be employed to determine whether there is an effect or significant differences on the observed values of the independent variables, for the subjects on account of thedifferent levels. It is a comparison of the meansat each level. The mechanism behind the comparison is however done on the basis of the deviation of estimated factor level mean from the grand mean which is the between group variation and the deviation of the observations around the factor level mean which is the within group variation. The within group variation accounts for the inherent randomness of the data whereas the between group variation accounts for both inherent randomness as well as any non-random variation due to some plausible external factors which indicate the differences among the levels (Roberts& Russo, 2014). It is these to variations which are compared to determine whether there is any significant external influence or not. Gliner et al. (2017), gives as example of a scenario where ANOVA is appropriate by referring to a study byStark-Wroblewski et al. (2006), assessing familiarity and interest of undergraduate students in five specific areas of professional psychology. The five levels were set to be, criminal profiling, clinical psychology, school psychology, counselling and forensic psychology. The interest of the students were quantified by an ordinal scale ranging from 0 which corresponds to no interest to 5 which corresponds to high interest. The response of the students were then recorded and the difference among the different levels were determined by using a single factor ANOVA method.

Mathematically, both ANOVA and linear regression are facets of a generalized linear model (GLM). A GLM is an ANOVA when all its predictors are categorical with a numeric response whereas it is linear regression when both response and sets of predictors are numerical(Fox, 2015).  The difference however lies in the utility of the two tools, that is, their scope of application as made apparent from the aforementioned examples.

Therefore, it can be said that although the two statistical techniques are derived from the same mathematical model under different specifications, they differ in the scope of application and relevance as well as the nature of the data

 

References

Fox, J. (2015). Applied regression analysis and generalized linear models. Sage Publications.

Gliner, J. A., Morgan, G. A., & Leech, N. L. (2017). Research methods in applied settings: An integrated approach to design and analysis. Routledge.

Lowry, R. (2014). Concepts and applications of inferential statistics.

Roberts, M., & Russo, R. (2014). A student's guide to analysis of variance. Routledge.

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