Under what circumstances,linear regression analysis is not adequate?

The simplest answer is where more than one factor impacts on the result data. And not all relationships with data are linear. Imagine ability to earn money realted to IQ - the correlation is likely to be non-linear because of the old addage it takes money to make money. Once you get past a level that can break the daily trudge, you're likely to get an exponential increase and then if you get to extremes of IQ it may well go down again (as the super intelligent dedicate lives to academia). OK not the greatest ro purest example but I hope you can see that wouldn't be linear!
When the correlation coefficient is low (less than 0.9), or when your phenomena is non linear (like exponential growth). Sometimes you can take logs and linear regression of that will work.
Ask this in the mathematics section.
Checklist. for determining whether linear regression is the right analysis for the data?

To check that linear regression is an appropriate analysis for these data, ask yourself these questions.

Question
Discussion

Can the relationship between X and Y be graphed as a straight line? In many experiments the relationship between X and Y is curved, making linear regression inappropriate. Either transform the data, or use a program (such as GraphPad Prism) that can perform nonlinear curve fitting.
Is the scatter of data around the line Gaussian (at least approximately)? Linear regression analysis assumes that the scatter is Gaussian.

Is the variability the same everywhere? Linear regression assumes that scatter of points around the best-fit line has the same standard deviation all along the curve. The assumption is violated if the points with high or low X values tend to be further from the best-fit line. The assumption that the standard deviation is the same everywhere is termed homoscedasticity.
Do you know the X values precisely? The linear regression model assumes that X values are exactly correct, and that experimental error or biological variability only affects the Y values. This is rarely the case, but it is sufficient to assume that any imprecision in measuring X is very small compared to the variability in Y.
Are the data points independent? Whether one point is above or below the line is a matter of chance, and does not influence whether another point is above or below the line.
Are the X and Y values intertwined? If the value of X is used to calculate Y (or the value of Y is used to calculate X) then linear regression calculations are invalid. One example is a Scatchard plot, where the Y value (bound/free) is calculated from the X value. See Avoid Scatchard, Lineweaver-Burk and similar transforms Another example would be a graph of midterm exam scores (X) vs. total course grades(Y). Since the midterm exam score is a component of the total course grade, linear regression is not valid for these data.

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