How do you know when a linear regression model is appropriate?
When it fits four assumptions : homogeneity, normality, fixed X and independence of the variables
Before applying your model
Checking for fixed X:
You should know the exact value of X before your analysis. In other words, the uncertainty on X has to be the lowest as possible. for example, you cannot take age as an explanatory variable if the lifespan is 25 years and you have an uncertainty of 3 years.
Checking for independence :
In the case of a multivariate linear regression, your explanatory variables have to be independent. In other words, do not use colinear variables in the same model.
To check this, plot one variable against the other. If you detect a strong linear or non linear pattern, they are dependent.
 Once you have applied your model
Checking for normality :
The residuals of your model (the variance not explained by your model) have to follow a normal distribution.
You can check this by an histogram of the residuals or by a quantilequantile plot.
You can see on the graphs below how it should looks like when you have normality.However, normality is not the most important assumption and linear models are robust enough to a small amount of nonnormality.
Checking for homogeneity:
This assumption is much more important. To check it, you can plot the residuals of your model against the fitted values.
You have homogeneity when the spread is more or less the same for all the residuals (you do not see any particular pattern, see figure below).If your residuals show a pattern (linear or non linear) or have a cone shape (spread higher in one side of the graph and lower at the other side), this assumption is not supported and you should find another kind of model.
You should do the same for each explanatory variable (X).
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A linear regression model is appropriate under the following conditions:

Linearity: There should be a linear relationship between the independent variable(s) and the dependent variable. This can be assessed through scatter plots or residual plots.

Homoscedasticity: The variance of the residuals should be constant across all levels of the independent variable(s). This can be checked through residual plots.

Independence of errors: The residuals should be independent of each other. This means that the error terms should not be correlated.

Normality of residuals: The residuals should follow a normal distribution. This can be assessed through histograms or QQ plots of the residuals.

No multicollinearity: If the regression model involves multiple independent variables, they should not be highly correlated with each other. Multicollinearity can lead to unstable parameter estimates.

The relationship should be theoretically plausible: The linear relationship between the independent and dependent variables should make sense based on the context of the data.
Check these conditions to determine whether a linear regression model is appropriate for the given dataset.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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