Edited:
In relation with our research, the respondents were deaf employees but due to the difficulty to have a large sample size, we're hoping to target that the respondents will be n=40 to reasonably say that it is a good estimate of statistical inference of the data for prediction using a simple linear regression analysis. And so, how many should be the participants to say that it is a good sample to have a good prediction? -like to have a good correlation, from my knowledge, (is it true? that n= 30 is the smallest good sample size) then, what about for a simple linear regression?
Another important consideration is the degree of variation in the X variable - more the better for detecting effects.
Say you want to know the effect of size of house (X) on houses price (Y); having all the sample as 3 bedroom properties would not work.- you would not be able to estimate the slope cofficient.
So think about sampling/selecting/designing the study to maximixe the variation in a key 'exposure' / predictor variable.
It depends on how you measure "goodness", and how variable (noisy) the data is. It also depends on the purpose of the analysis. If you want a single-case prediction, the prediction interval (the width of the prediction band) is of interest. This is mainly depending on the noise (variance) of the data, and all you need is enough data to get a reasonably good estimate of this variance. If you want an estimate of the expected value (or of the slope of the regression line), the confidence interval of the slope coefficient is of interest. This interval also depends on the noise of the data, but it will become narrower with more data.
A reasonable rule of thumb is 5-10 observations per IV. If alpha and power are the important things then calculate a sample size as Prof Wilhelm suggested. It matters some whether you have an explanatory or a predictive model. For a predictive model consider the prediction interval while for an explanatory model consider the term that gives the largest n. See the link. Best wishes, David Booth
Google To E xplain or To predict Shmueli
Sample size determination is a big topic, I believe it is difficult to find any rule-of-thumb answer that has a reasonable amount of validity. Essentially, it depends on what analysis you want to do and how good the result you want to achieve. Material in the attached file, including those in its reference, may be a good starting point to you. You can also find this document via the following link:
http://www2.sas.com/proceedings/sugi26/p240-26.pd
Practically there is no rule of thumb for sample size in regression. However, few literature suggested a figure like 'number of variables * 40 '
Another important consideration is the degree of variation in the X variable - more the better for detecting effects.
Say you want to know the effect of size of house (X) on houses price (Y); having all the sample as 3 bedroom properties would not work.- you would not be able to estimate the slope cofficient.
So think about sampling/selecting/designing the study to maximixe the variation in a key 'exposure' / predictor variable.
For linear regression, the rule-of-thumb is 10K+50 (K is the number of Independent Variable) will be better to predict.
Another is based on Nunnally, 60 sample is good for the regression analysis,
It is also important to consider how correlated the exposure (the explanatory variable of key interest) is with the 'control' variables - see attached - in any practical application this is important ; NQuery was used to do the calculations. Simple rule of n time m predictor variables do not take this into account and yet the whole point of doing multiple regression is to condition on these other correlated predictors.
I generally agree with Jochen, but will make one comment. Yes the variance of the slope b becomes smaller with increased n. For the case of the classical ratio estimator, where sigma is for the random factors of the estimated residuals, the estimated variance of b, used in the confidence interval of the slope, b, is as in the attached figure. As n becomes larger, the sum of those x-values becomes larger, and variance of b is reduced. However, this is also true of the prediction interval, for any given x, as the attached variance of the prediction error also becomes smaller with more data, as shown in this example.
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