In your situation, as I understand from your description, the question you intend to or indeed can only answer, given the binary nature of  the X variable, is the difference in Y when X= 0 and X=1. Without bothering about the linearity that you are referring to, just fit the linear regression 

Y = a + b X 

The estimate of the slope 'b' provides the answer to the question. However, before accepting the results from the regression analysis, make sure that the residuals from regression analysis have homogeneous variance and a normal distribution. 

Linearity can only be tested if we have at least three points. I assume you are therefore trying to test whether the assumption of linearity of the association between the continuous variable and your independent variable is met. 

Why favour entering a dependent variable as a continuous variable rather than entering a dichotomised variable?

If an association is dose dependant, conserving information on the value of a measured factor will improve our ability to model it's association to the independent variable. In other words, the analysis will also test whether the association is "dose-dependent" and thereby provide support towards a causal relationship.

Why not just create dummy variables for different ranges of values instead?

This approach indeed has the advantage of not having to assume that the magnitude of the association is always proportional to the value of the dependent variable. It however has a major disadvantage! When creating ranges of values, we generate what we call dummy variables. Each new variable then corresponds to one of the ranges (n= number of ranges). Therefore, we are using n-1 degree of freedom in the model. This considerably reduces the power of the overall analysis especially if some ranges only contain very few participants. For this reason, we favour testing the assumption of linearity.

What are we testing?

When entering a continuous variable in logistic regression, the reported indice from the logistic regression corresponds to the average logged odd ratio for each unit of the independent variable. For this to make sense, it is necessary to verify that it remains more or less constant over the entire scale.

Generating dummy variables

This can be done by first transforming the continuous variable into an ordinal one. The number of ranges should depend of the sample size. The more the better, but each range should contain enough participants to have about 10 participants with the outcome of interest. First construct the ranges with similar paces between each range. 

Chose the reference value and code it dummy_0. Code the other ranges by units from then (negatively bellow and positively above). Compute the logistic regression model entering each range as a separate variable (keep out the reference variable). Visually analyse the magnitude of association of each variable and check if it seems linear. If this is not the case, consider transformation and start the operation again. 

Testing linearity – likelihood ratio test

The idea is to see whether we have lost information when considering the association to be linear. Therefore, what we do is to compare the model with dummy variables with the model with the same measure entered as a continuous variable. If the models significantly differ, this means we cannot assume the association to be linear. Therefore, we perform a likelihood ratio test between models and if p>0.05 then we can state that absence of linearity was not observed.

On STATA

Stata automatically generates dummy variables using the xi: command.

gen dependant2=dependant // generates a new variable

recode dependent2 0/0.999=0 1/1.9999=1 2/2.9999=2 3/3.9999=3 4/4.9999=4 

xi: logit independant i.dependant2, or

estore A

xi: logit independant dependant, or

estore B

lrtest A B

Hi,

Thanks for the answer. To be more clear, my dependent variable is a score which is determined based on presence or absence of an error. Though there can be many & different types of errors so all the independent variables are 1/0. Whereas based on presence of error(s) the score varies, more errors could mean low score or a specific type of error could mean not so low score in comparison to another error.

Thus, my score (dependent variable) has to be a continuous value (1-10).

Now if I need to determine whether the X's & Y's are linear then how do I decide as I had already mentioned that X's will be only 0/1 whereas the score (1-10).

See if these links may possibly help to you to ask the 'right' question

http://www.theanalysisfactor.com/the-distribution-of-independent-variables-in-regression-models-2/

http://stats.stackexchange.com/questions/118051/how-to-perform-residual-analysis-for-binary-dichotomous-independent-predictors-i

Here is already some discussion on this issue https://www.researchgate.net/post/Which_test_do_I_use_to_estimate_the_correlation_between_an_independent_categorical_variable_and_a_dependent_continuous_variable

Since in correlation there is no truly dependent and independent variable, you can switch from one to another at will, why don't you use logistic regression and your dichotomized variable becomes your outcome variable and your continuous variable becomes your predictor? That solves that problem. In this case, you would have the odds of obtaining either yes/no according to different values of the continuous variable. Since I don't know what is one and what is the other, I cannot say if this would make sense, but it is a practical way of circumventing the problem.

@ chandra Thanks a lot for the links. They were indeed very helpful.

Thanks Gloriam.

Performing a linear regression with a dependent (response) variable on an independent binary variable is equivalent as doing a 1 way ANOVA. 

if the IV is dichotomous (0, 1) and DV is continious, it is accepted in Research and statistics books to run linear regression and Point biserial correlation. The assumption for the linearity behind the dichotomous (which is exactly as if you are doing Dummy coding) that the line is drawn between the 2 mid points in the 2 groups.

@ Prof. Ahmad: I have single DV (score) but multiple IVs (errors) and each IV is dichotomous (0/1). Will I be required to perform a biserial correlation for each IV and the DV individually or in one go (say using SPSS).

@ Jean: As mentioned above I have multiple IVs (different types of errors) will 1 way ANOVA still be useful or I need to use 2-way ANOVA.

Another point I have the DV i.e the scores for different sentences (nearly 30) based on the different types of errors present/ absent in the sentence. Also all these sentences have been scored by 30 individuals. 

How do I form the groups?

I think first you test the probability distribution of the Dependent variable before using linear regression for the analysis purposes.

After coding a bivariate categorical variable, the relationship is by definition linear.  There's no reason to check for satisfying the assumption in that case.  You only have two points to connect.  

As an aside, if your two values were actually two values along a continuum (e.g., two dosages), then you still couldn't test for linearity.  However, the relationship in this case could be nonlinear but you wouldn't know unless you sampled more values along the continuum.  

There are two factor only, the outcome is continuous, the predictor is binary, hence you are comparing the difference of two means.

You do not need linear regression; you should use t-test. The assumption is the dependent variable is normal distributed for each group (absent or not). If the variance for each group is the same, you can use the classic t-test, if not you need use Welch–Satterthwaite equation to evaluate the degree freedom. I think many software will do it for you.

If you use linear regression, the same result for mena difference and significance will be get as t-test by assuming equal variance.

If the dependent vaiable is not normal distributed, you may do transformation, then use t-test or use non-parametric test for the medians.

Thanks a lot all of you for the response.

Hello, what are you exactly trying to verify? Do you want to know if after a certain value of the continue variable, the probability that the categorical will  have the value 1 increases dramatically, and that after a certain value of the continues variable the categorical value will attain always the value 1? Check if this is exactly what your are trying to find out. Such type of problem requires a fit of the categorical character (it will be the dependent variable) to the continue variable (the independent variable) by a non linear model. There are, for example the Probit model and the logit model, the latter is a logistic regression. You may use the logistic regression  in a problem where, for example, you try to find out the size at first sexual maturation, or the size at sex change, or the size when certain exploited resources reach market size. Usually the value L50 of the continue (independent variable) is of interest, This value represent the point at which   50% of your population, whose density distribution are fitted by the logistic model,   attain the categorical value 1. The logistic regression model obtains a the function and from this funcion you can calculate the required L50. The analysisof the logistic regression use various models to estimate the lost function, which are needed for the fit. SPSS and Statistica software package allow you to make this type of analysis. I suppose R also, but this programm is less user friendly. I hope it helps. Best regards