I'm not sure why there's any discussion here. The equation of a straight line is y = mx + c. This is an inviolate mathematical formula. The equation itself justifies nothing - it's perfect. Data sets are fitted with an appropriate model which in the case of a linear fit is usually a least squares procedure. The equation itself tells you nothing about the data going into that best fitting procedure. The 'goodness' of the model is assessed by deviations from that model. Here correlation coefficients (e.g. R2) can be employed. So, the equation on its own tells you nothing about whether that model is appropriate or not. How close theory and practice are, however, tells you something about the appropriateness of the model. Be careful with 2 things:

• Inappropriate correlations. This is where an understanding of the basis of the model is so important. I've shown a couple above
• Over-complicating the situation. n points on a plot can be fitted perfectly by an equation of (n - 1)th degree. This definitely does not mean that the more complicated model explains more. On the contrary, it's basically an empirical fitting...

@Hesam Amou It's not 'better' - it's mandatory to show the data that gave rise to the best fit

Dear Srinivas Lakkoju ,

to my opinion it is not sufficient to claim linearity only by presenting a linear equation without showing the graph combined with the data.

You might fit a linear function onto data which by far do not show linearity by eyes. From the mathematical point of view you may fit anything by anything.

So to convince the reader I would prefer to show the linear fit combined with the data. Saving space is not an argument to hold back the graph.

But it is just my personal point of view....

you can test the linearity of your model and report the result.

Srinivas Lakkoju,

If you're wanting to test whether or not there's a linear trend between two continuous variables, you could try using Pearson's correlation coefficient (assuming you meet the assumptions). This would result in a single value (+/- a confidence interval and/or a p-value) summarizing the strength of a linear trend between the two variables.

However, this approach or any other approach that only looks for straight-line relationships can very easily miss meaningful non-linear trends between the two variables (think of fitting the least squares straight line through a parabola). This, in my mind, is why reporting just a Pearson's correlation coefficient or just the coefficients from a linear regression equation is not sufficient. If the linearity of this trend is important to the study, then I would recommend either including a scatterplot along with the least-squares regression line or providing some other sort of numerical evidence that the trend is unlikely to be non-linear (there are several ways of doing this via regression modeling).

No it is not sufficient

Dear Gerhard Martens, Niyungeko Antoine, Niyungeko Antoine and Joan Nyika,

Thank you for your quick and useful clarification. Now, I understand that I have to show the scatter plot with equation.

Thank you.

Srinivas Lakkoju

Definitely not as all statistical textbooks will tell you. Take a look at:

https://www.graphpad.com/support/faq/smoothing-data-can-make-it-seem-as-though-bad-predictions-forecasts-are-quite-accurate/

I have attached Figure 1 from this. The points have been put on the plot with a shotgun..

Srinivas Lakkoju

Also look at:

https://en.wikipedia.org/wiki/Correlation_and_dependence

Srinivas Lakkoju

And...

https://www.tylervigen.com/spurious-correlations

That's a good question

No.

It's better to be presented together

I'm not sure why there's any discussion here. The equation of a straight line is y = mx + c. This is an inviolate mathematical formula. The equation itself justifies nothing - it's perfect. Data sets are fitted with an appropriate model which in the case of a linear fit is usually a least squares procedure. The equation itself tells you nothing about the data going into that best fitting procedure. The 'goodness' of the model is assessed by deviations from that model. Here correlation coefficients (e.g. R2) can be employed. So, the equation on its own tells you nothing about whether that model is appropriate or not. How close theory and practice are, however, tells you something about the appropriateness of the model. Be careful with 2 things:

• Inappropriate correlations. This is where an understanding of the basis of the model is so important. I've shown a couple above
• Over-complicating the situation. n points on a plot can be fitted perfectly by an equation of (n - 1)th degree. This definitely does not mean that the more complicated model explains more. On the contrary, it's basically an empirical fitting...

@Hesam Amou It's not 'better' - it's mandatory to show the data that gave rise to the best fit

Dear Srinivas Lakkoju, if it's really necessary to save space in your manuscript, you can present some of the scatter plots as Supplementary Information (SI) and refer to them in the main text.

@Alan F Rawle , I think you didn't understand Srinivas Lakkoju question,

Hesam Amou

I understood the question perfectly. The y = mx + c equation does not 'prove' linearity.

The problem here is that the linearity assumption means linearity in the regression coefficients, NOT Y vs X. Please read any reputable book on linear Statistical models. My fav is: https://www.amazon.com/Applied-Linear-Statistical-Models-Michael/dp/007310874X/ref=sr_1_1?crid=NHHB65RM7664&dchild=1&keywords=kutner+applied+linear+statistical+models&qid=1590764938&s=books&sprefix=applied+linear++models+kutner%2Cstripbooks%2C218&sr=1-1

Best wishes, David Booth, ASA PSTAT

David Eugene Booth

Thank you for your explanation. One handy hint to reduce the size of url's is to use tinyurl.com. This takes your huge url and converts it to a forever 7-character alphanumeric. In this case: https://tinyurl.com/y8wxhfqu

It's an excellent book, BTW. Less expensive earlier editions (e.g. $5) can be found on Abebooks: https://tinyurl.com/ybpl5dvm

Alan F Rawle Thank you very much. David Booth

Srinivas Lakkoju

Is straight line equation sufficient to claim linearity, without a graph?

The answer is YES.

Regards

No. But if the concern is solely for publication purpose, the linearity could be stated in the text while the plots are provided as attachments. Except the main reason for the publication is to show this relationship between variables.

The more transparent your work is, the more convincing, even for yourself. So it is better to include the key elements of analysis with which you reached your conclusions. As time goes by, you will appreciate being able to read again without regretting having left things unclear.

My response is no for obvious reason:

As a forester, I know that the ideal relationship between tree height and diameter is nonlinear with a kind of sigmoidal properties of monotonic increment, inflection point and asymptote. But some authors have ignorantly used simple linear equation to represent this relationship; which is not biologically reasonable.

So, I think a simple scatter plot of X vs Y should be the first point of call before fitting your equation except there is a prior knowledge of the relationship between the variables.

Hello

You should mention that what kind of relationship is exsit between your variables. Because It is different for same species in different regions.

Plotting and showing the plots of the data lets me feel safe that the response (whatever) is linearly related to the variable. To be only convinced with equation will prevent us from seeing the real behavior of the many cases. However, getting a linear relation is a real achievement especially when we deal with natural properties and simply with standard solutions prepared in a well furnished laboratories. Linear relation (plot) between certain property for natural samples will show you the minor influence of other parameters in the natural sample on that property.

Srinivas Lakkoju A straight line (and its equation, y = mx + c) is linear by definition (needs 2 points only) and proves nothing. You can claim whatever you want (e.g. the moon is made of cheese), but the raw scatter data and parameters such as R2 and fit parameters (least squares) will justify the selection of such a (linear) model in preference to others.

Is the linearity critical to the conclusions of your research? If so, you need to include the graph. If not, it may raise questions of transparency and laziness. Find something else to remove/shorten. Alternatively, draft the paper with all the graphs you would ideally include and have people review it.

I do agree with Dr. Alan F Rawle

I see some responses from people I admire for their experience and ideas. I only think the following basic to keep in mind. The most important for a regression model is the substantive knowledge of the subject. If this knowledge gives us insights into the mathematical expression of the model, and especially if the parameters have meaning in the researcher's work environment, we should not hesitate to consider it to model the data. If we find a mismatch, we must very thoroughly explore explanations and not stop with the easy solution of looking for a model that better fits the data. Here R2 comparison could be a wrong and dangerous guide. If feasible, it is best to take learning and validation samples to examine the potentiality of the models to be generalized consistent with the theory. In other cases, without a guiding theoretical model, we only have the predictive option, where the substantive meaning is speculative, with the adequate justification the researcher may find. Here, adjustment indicators can play a better role.

Thanks for the valuable insights and clarifications. I understand that I have to compulsorily present scatter plots (graphs) to claim linearity. Of course, scatter plots produced in SPSS also include mathematical equation of the related straight line.

Dear researchers, I am indebted to all of you. Thanks.

Dear Srinivas Lakkoju

For connoisseurs of mathematics and statistics, simply see the results in solving the equation of the straight line; It gives you an idea in your mind of whether it really expresses linearity (value of the slope, the correspondence of the values ​​of X and Y; and the value of the intercept), but it is necessary, that it be expressed graphically, in this way our senses and thoughts complete the mental image in a sensible and tangible way in a plane or two dimensions.

regards

José Luis

I agree completely with Alan and do hope you would add the scatter plots in order to avoid the rejection of your paper.

When you use linear regression, you inherently assume that the relationship is linear. To check for this assumption if you only have one variable then a scatterplot as suggested above should suffice. This can always be provided as supplemental material. However, when you have more than one explanatory variables in your model then you should consider techniques such as component plus residual plots.

For the appropriateness of your model, you should also look at residuals, influential points, and collinearity.

I hope this helps

Linear in linear models Means linear in the regression coefficients. Just up in a goo linear models book. D. Booth

yes linearity must also occur i regression coefficient's in addition to the graphical picture.

Ceyhun you are incorrect otherwise how are glms like logistic regression linear. Read a linear models book Like Kutner et et al Applied Linear Statistical models. This is about linear algebra not Y-X scatterplots. D. Booth.