Once you have fitted a linear model using the regression analysis, you will need to determine how well the model fits the data. Those coefficients (R, R^2,adjusted R) quantify the 'model quality', or the proportion of the results variance that can be explained by the model. Have a look at this site: 

http://blog.minitab.com/blog/adventures-in-statistics/regression-analysis-how-do-i-interpret-r-squared-and-assess-the-goodness-of-fit

If you are using a multiple linear regression, you need to look at the R^2 (adj). I would also look at R^2 (pred). R^2 (adj) tells you what percent of the total variability is accounted for by your model. If R^2(adj) is 0.6897, then your model accounts for 68.97% of the total variability.

What is your VIF for each term in your model. That is more important, initially, than an R^2 value.   

In multiple regression analysis the "Adjusted R squared" gives an idea of how the model generalises. In an ideal situation, it is preferable that its value is as close as possible to the value of "R squared". So, the proportion of the variance explained by the multiple regression model is indicated by "R squared". If a hierarchical regression has been conducted, then then the improvment of the model can be assessed at each stage of the analysis by looking at changes in "R squared" and assessing the significance of such change.

There is the assumption that errors in regression are independent. This assumption can be met if the Durbin-Watson statistic is around 2 (and between 1 and 3) 

A multiple regression model is given in

Y. Cheng, The Response Surface Methology, M.Sc. thesis, Indiana University of South Bend, 2007

https://www.iusb.edu/math-compsci/_prior-thesis/NBradley_thesis.pdf

See Section 4.1.1, starting on page 7.

Simply put, R is the correlation between the predicted values and the observed values of Y. R square is the square of this coefficient and indicates the percentage of variation explained by your regression line out of the total variation. This value tends to increase as you include additional predictors in the model. Thus, one can artificially get higher  R square by increasing the number of Xs in the model. To penalize this effect, adjusted R square is used. When you compare models with their complexity, you should then rely on Adj R square. Predicted R square is another measure which addresses the issue of overfitting the data  and explain the prediction power for future observations. 

Due to multicollinearity, the coefficients on individual variables may be insignificant when the regression as a whole is significant. Under this condition, It is a good idea to look at the P value for the regression as a whole. It tells how confident you can be that each individual variable has some correlation with the dependent variable, which is the important thing.

R^2 is the proportion of sample variance explained by predictors in the model. Thus it is the ratio of the explained sums of squares to the total sums of squares in the sample. R is the multiple correlation coefficient obtained by correlating the predicted data (y-hat) and observed data (y). Squaring R gives you R^2. Thus R^2 is a function of the quality of prediction within the sample.

Adjusted R^2 attempts to generalize the R^2 statistic as a population estimate. Using R^2 as a population estimate leads to bias in small samples and this increases with more predictors.

I find R^2 more useful as a descriptive quantity (albeit not very useful) and the difference between R^2 and adjusted R^2 is a helpful reminder that the model is likely to over-fit. I don't find adjusted R^2 that useful because I think the population value of R^2 is generally not a useful quantity per se.

Dear Rahman

In the simple regression, the R - square and adjusted R-square are same

But in the multiple regression, You have to care about Adjusted R - square and the difference between R - square and Adjusted R - square. If you put one more varibale into your model, but the adjusted R - square is not increased or even it decreases, you have to consider the suitable of this variable in your model

BEst

Phuc Canh

In lay man, language  (Statistics) tell what is R and R-square

Dear Respected colleague,

In order to assess the overall regression model fit in supporting the research hypotheses. This is done by, firstly, examining the adjusted R squared (R2) to see the percentage of total variance of the dependent variables explained by the regression model. Whereas R2 tell us how much variation in the dependent variable is accounted for by the regression model, the adjusted value tells us how much variance in the dependent variable would be accounted for if the model had been derived from the population from which the sample was taken. Specifically, it reflects the goodness of fit of the model to the population taking into account the sample size and the number of predictors used. Researchers suggests that this value must be equal to or greater than 0.19.

I hope this answer your question,

Best Wishes,

Well illustrated by Anteneh Ayanso and Martín Martínez Villar I fully agree with them

R: It is the correlation between the observed values ​​Y and the predicted values ​​Ŷ.

R2: It is the Coefficient of Determination or the Coefficient of Multiple Determination for multiple regression.

It varies between 0 and 1 (0 and 100%), sometimes expressed in percentage terms.

In this case, it expresses the amount of variance in the data that is explained by the linear model.

Thus, the higher the R2, the more explanatory the linear model is, that is, the better it fits the sample.

It is a statistical measure of how close the data is to the adjusted regression line.

It is the percentage of the variation of the response variable that is explained by a linear model.

R2 = Explained variation / Total variation

R2 = 0% Indicates that the model does not explain anything about the variability of response data around its mean.

R2 = 100% Indicates that the model explains all the variability of the response data around its average.

For example, R2 = 0.8234 means that the linear model explains 82.34% of the variance

of the dependent variable from the regressors (independent variables) included in the linear model.

R2 adjusted:

The inclusion of numerous variables, even though they have very little explanatory power on

the dependent variable, will increase the value of R2.

This encourages the indiscriminate inclusion of variables, undermining the principle of parsimony

(see more widely in Ockhan's razor).

To combat this trend, use adjusted R² as an alternative measure to R2,

penalizing the inclusion of little explanatory regressors.

This will be helpful https://www.reneshbedre.com/blog/multiple-linear-regression.html

Can we say that R = sqrt(R^2) ?

Hi Mohamed Nedal

You can say that R = sqrt(R^2), but you should not interpret it that way. In the case of multiple regression analysis, R is the Multiple correlation coefficient, and it is the square root of the R-Squared. But, here if you are referring to the standard correlation coefficient (r) between two variables, then the equation will not be valid as r range from -1 to 1. Read more here https://www.reneshbedre.com/blog/linear-regression.html

Thank you Renesh Bedre

Can we say that R = sqrt(R^2) ?

Do not forget, that R might be also negative, it has values from -1 to 1. Just from R^2 you cannot say if the R was positive or negative. So no, we can not say that.

For any model when we find the values of R and R2, the former is square of the later. Question is why we square it, R may be positive or negative showing the direction of relationship, and R2 makes it positive showing only the strength of relationship, but it also decreases the value somewhat. So if we just want to show the strength of relationship without sign, we could have taken the mod of R, why square, this is the question that I could not find the answer about. May be there is no reason to find the relationship between R and R2.

R squared, when regarding a linear regression or correlation best fit trend line/the linear equation (one in the form of Y = aX +b… where a and b are constants and can be positive or negative) has the cool effect of also telling you the percentage of total variation in Y explained by the variable X… basically, if you know what X is (say X is the maximum wind speed during the prior 12 hours at everest base camp or X is Mean Arterial Pressure times hematocrit at rest while at camp 2 during a summit push) and Y is visibility or precipitation in the following 12 hours in scenario one and the second is the percentage of total circulating fluid volume escaping pulmonary or cerebral capillaries relative to the amount reabsorbed. say the R is 1… means it predicts it perfectly, wind speed of x means over the next 12 hours visibility will be y feet… no variation… and if the relationship between your blood pressure and hematocrit is X than you’ll be suffering from high altitude pulmonary and cerebral edema that is causing swelling exactly as fast as the estimate provided by that variable. but what if it’s an R of .7. It almost feels misleading to have a number so close to perfect when it basically accounts for only 49% of the variability in Y. If the Y variable were mortality rate in people climbing during those circumstances than depending on the person it goes from “so even if I had the best hematocrit/blood pressure coefficient more than half of what will determine if I live or die from pulmonary edema is still essentially coming down to chance from interpreting it as R= .7 and I got the minimum X value, that means even if every other variable aligns against me I’ll still be fine 70% of the time and what are the chances everything else goes wrong anyway, I'm guaranteed to be fine”. so

we see R2 more unless we want to know direction because it’s more meaningful and easy to grasp as a measurement of predictive power.

Yes