So you have a sample of responses for 3 parallel versions of questionnaire, where some of the questions between versions are the same and you have one version where all the questions are present, correct?

For me, this sounds like an test equating problem. Consider that while you want the tests to be parallel, you do not know if people answered all the questions the same way. So there could be different response patterns for question 1 and question 4 and if so, then the same raw sum of scores on two tests consisting of different questions would mean different things. In this case you could employ test equating, that is, a statistical procedure that lets you to transform raw scores of one questionnaire into the scale of another, so that both tests share the same scale and are equivalent.

The simple example is linear equating, where if you want co transform raw score of questionnaire $X$ into scale of $Y$ you use mean and standard deviation:

$$Lin_Y(x) = \frac{\sigma_Y}{\sigma_X}x + \left( \mu_Y - \frac{\sigma_Y}{\sigma_X}\mu_X \right) $$

there is also more advanced method that use cumulative distribution functions - the equipercentile equating method:

$$Equi_Y(x) = F^{-1}_Y \left[ F^{-1}_X(x) \right]$$

However, the second method needs special data preparation beforehand i.e. continuization of the scores. Of course, there are also other methods as well.

Wile the questionnaires were answered by different populations, you could use the questions that are common between versions of questionnaires as anchor tests in equivalent groups design.

Check books on equating by Kolen and Brennan and von Davier at al for more informations.

So you have a sample of responses for 3 parallel versions of questionnaire, where some of the questions between versions are the same and you have one version where all the questions are present, correct?

For me, this sounds like an test equating problem. Consider that while you want the tests to be parallel, you do not know if people answered all the questions the same way. So there could be different response patterns for question 1 and question 4 and if so, then the same raw sum of scores on two tests consisting of different questions would mean different things. In this case you could employ test equating, that is, a statistical procedure that lets you to transform raw scores of one questionnaire into the scale of another, so that both tests share the same scale and are equivalent.

The simple example is linear equating, where if you want co transform raw score of questionnaire $X$ into scale of $Y$ you use mean and standard deviation:

$$Lin_Y(x) = \frac{\sigma_Y}{\sigma_X}x + \left( \mu_Y - \frac{\sigma_Y}{\sigma_X}\mu_X \right) $$

there is also more advanced method that use cumulative distribution functions - the equipercentile equating method:

$$Equi_Y(x) = F^{-1}_Y \left[ F_X(x) \right]$$

However, the second method needs special data preparation beforehand i.e. continuization of the scores. Of course, there are also other methods as well.

Wile the questionnaires were answered by different populations, you could use the questions that are common between versions of questionnaires as anchor tests in equivalent groups design.

Check books on equating by Kolen and Brennan and von Davier at al for more informations.

So you have a sample of responses for 3 parallel versions of questionnaire, where some of the questions between versions are the same and you have one version where all the questions are present, correct?

For me, this sounds like an test equating problem. Consider that while you want the tests to be parallel, you do not know if people answered all the questions the same way. So there could be different response patterns for question 1 and question 4 and if so, then the same raw sum of scores on two tests consisting of different questions would mean different things. In this case you could employ test equating, that is, a statistical procedure that lets you to transform raw scores of one questionnaire into the scale of another, so that both tests share the same scale and are equivalent.

The simple example is linear equating, where if you want co transform raw score of questionnaire $X$ into scale of $Y$ you use mean and standard deviation:

$$Lin_Y(x) = \frac{\sigma_Y}{\sigma_X}x + \left( \mu_Y - \frac{\sigma_Y}{\sigma_X}\mu_X \right) $$

there is also more advanced method that use cumulative distribution functions - the equipercentile equating method:

$$Equi_Y(x) = F^{-1}_Y \left[ F^{-1}_X(x) \right]$$

However, the second method needs special data preparation beforehand i.e. continuization of the scores.

Wile the questionnaires were answered by different populations, you could use the questions that are common between versions of questionnaires as anchor tests in equivalent groups design.

Check books on equating by Kolen and Brennan and von Davier at al for more informations.

So you have a sample of responses for 3 parallel versions of questionnaire, where some of the questions between versions are the same and you have one version where all the questions are present, correct?

For me, this sounds like an test equating problem. Consider that while you want the tests to be parallel, you do not know if people answered all the questions the same way. So there could be different response patterns for question 1 and question 4 and if so, then the same raw sum of scores on two tests consisting of different questions would mean different things. In this case you could employ test equating, that is, a statistical procedure that lets you to transform raw scores of one questionnaire into the scale of another, so that both tests share the same scale and are equivalent.

The simple example is linear equating, where if you want co transform raw score of questionnaire $X$ into scale of $Y$ you use mean and standard deviation:

$$Lin_Y(x) = \frac{\sigma_Y}{\sigma_X}x + \left( \mu_Y - \frac{\sigma_Y}{\sigma_X}\mu_X \right) $$

there is also more advanced method that use cumulative distribution functions - the equipercentile equating method:

$$Equi_Y(x) = F^{-1}_Y \left[ F^{-1}_X(x) \right]$$

However, the second method needs special data preparation beforehand i.e. continuization of the scores. Of course, there are also other methods as well.

Wile the questionnaires were answered by different populations, you could use the questions that are common between versions of questionnaires as anchor tests in equivalent groups design.

Check books on equating by Kolen and Brennan and von Davier at al for more informations.