Dear Melanie, Can you please explain the following:

1. Do you have  any observation with all three variables measured?

2. What do you call beta?  Is this the square of the correlation, the absolute value of the correlation, the correlation  itself or something else?

The first question is important, since in case of NO your problem will have non-unique solution. The second is just for setting right further discussions:)

Best regards, Joachim

Hello Joachim,

1) I conducted a survey with 176 people, answering all questions related to the variables. However to do the regression analysis I split the genders into f & m, which were distributed unequally. So I reduced the number of female to match the amount of male participants. So now I did a regression using all participants to find the relation between X and Y and another two to measure the impact of females and males. Females had a higher impact, and now I need to calculate the final impact of X and X2 together on Y. -- I hope this explains my situation a bit better and answers your first question.

2) beta is for me the unstandardized beta coefficient in the regression analysis, which is practically the same a R (not R squared)

Thanks in advance for your help!!

Best Regards, Melanie

Use the most recent version of SPSS from IBM to do this task.

Dear Melanie,

Thanks for your explanation. It shows, that the situation is different than my first understanding. However, still it is not clear for me what is the structure of your data. It is very important to determine clearly what are the relations between variables which you call independent, dependent and what the gender is in your research. Is this another variable? Obviously, it would be sufficient to present the variables in symbols, only. Now I guess your data have the following structure:

1.A  single observation consists of three variables  X1, X2  and  Y;  Y is considered as dependent on  both,  X1  and  X2. GENDER IS AN ADDITIONAL VARIABLE.

2.The sample is of size 176, among them:    n1  of gender  f  and  (176 - n1) of gender  m

 If this is correct, then two questions are basic for further understanding:

What is 30?  What impact is described by the two beta coefficients?

Possibly, I am misunderstanding the situation. Then the questions are meaningless, of course.

Best Regards, Joachim

Hello Joachim,

In my case gender is the variable X2. So X1 has an impact on Y with n=176, which is the beta coefficient of 0,465. Among my sample I have female=146 and male=30. To assess the impact of female and male fairly I reduced female to 30 to have an equal amount of male and female cases and did another regression between X1 and Y using only 30 female cases, and another also between X1 and Y using only 30 male cases. From these two regression females showed a higher impact with beta coefficient of 0,733.

So to calculate the final impact on Y I need to consider the total impact of X1, but also the impact of females. 

OK, Thanks, now I am much more close to full understanding. The only question remains, what is the beta coefficient for the impact of x1 within males participants. Why are you omitting it?  A appretiate your patiency in answering this possibly non-applicable questions. Joachim (with best regards, as usually;)

Hello Joachim,

the beta coefficient for males is 0,655, it is also significant, but females have a higher impact that's why I chose to omit males.

Regards

Hello Melanie,

Thanks for quick answer. With respect to omitting male's factor you are definitely wrong. Sorry, but if you wish possess the "total impact"  (seemingly - of males and females), you cannot omit the worst part of people (smile). Now moreserious details.

If you you have two populations with two observable quantities, say  X  and  Y, it is a good way to name them separatly for each population, say  X1, Y1   and  X2,Y2   for female and male gender, respectively. What you call total impact, to the best of my understanding, is the correlation of the random variables  X  and  Y  in the MIXTURE  of the two populations. If you agree with this point, I can provide you some formulas which calculate the correlation of the mixture in terms of the parts.  

Thus, the first step is to find all moments of order 1 and 2  within each part separately, without reducing them to equal size. THIS IS VERY IMPORTANT FOR BETTER USE OF ALL DATA. Let us call them   ml(j,k),  for  (j,k) = (1,0), (0,1), (2,0),(1,1), (0,2), where  l=1  corresponds to the femle subsample, and  l=2  to the male part.  Thus, e.g.    m1(2,0)  stands for the second moment of  X   in the female subsample (of 146 persons).   With these you can also find   the central moments of order  2  denoted by   \mul(j,k)  with  (j,k) = (2,0)  (the variance of  X), (j,k)=(1,1), the covariance of  X  and  Y)  and  (j,k)=(0,2) (i.e. the variance of  Y).  All these parameters have to be calculated by use of the data, SEPARATELY  for  females (l=1) and for males (l=2, consisting of 30 members).

The next step, is to find somewhere in  statistics of the population investigated (like in government statiscical offices)  the fractions of   female  and male members of the modelled community.  Let us call them   p1  and  p2, respectively. Using these and the above numbers, you can find the parameters of the mixed population according to the following formulas, valid for the non-central moments, only:

(0)    m(j,k) =   p1  m1(j,k) + p2 m2(j,k)     is the moment of order (j,k) in the mixture.

Now you can use for  the correltion between  X  and  Y  in the mixed population the standard  final formula 

(1)  \rho = {  m(1,1) - m(1,0) m(0,1) } /  \sqrt{  (m(2,0) - ( m(1,0) )2 ) ( m(0,2) - (m(0,1) )2 }

It is not a simple dependence on  \rho1  and  \rho2 !   For instance, if both  mean values  of female and male population can be  accepted as equal, the final formula can be simplified  to the following

(2)                               \rhoX,Y = LICZNIK / MIANOWNIK,

where

LICZNIK = p1 \sqrt{ vX1  vY1 }  \rhoX1,Y1  +   p2 \sqrt{ vX2 vY2 }

MIANOWNIK = \sqrt{  vX  vY }

and  the   v-s   are variances of the indicated variables within the separated  and the mixed populations  and  \rho-s  are the impact factors in the mixed population (the LHS)  and in the separated populations  (on the RHS).

IMPORTANT:  The last formula (2)  works only in case of  equal  means

m1(1,0)  =  m2(1,0)      AND    m1(0,1) = m2(0,1) .

Hopefully, my advise will help.

Best wishes, Joachim

PS.  The above formulas are all spread over the handbooks on probability, however I am not able in short time find concrete references. JoD

PS2. The last formula is the simplest posssible, and is does not take into account the different sizes. The influence of difference can be diminished by suitable (I do not know how) choice of weights i.e. coefficients in formula (0).  One of possibilities is replacing  (0)  by the following:

(0')    m(j,k) =  [ p1 n1 m1(j,k)  +  p2  n2  m2(j,k) ] / [ n1 + n2]

where  n1  and  n2  in the considered case equa  146  ans 30, respectively.  This proposal is not justified by a strictly formulated reasn. Juast a guess:)

Joachim

Hello Joachim,

thanks a lot for this detailed answer!! Unfortunately I am not good in statistics..

What exactly do you mean with order 1 and 2 and how can I calculate (j,k)=(1,0), (0,1), (2,0),(1,1), (0,2)?

Is there maybe a way to do this with SPSS? This would be much easier for me.

Thank you so much!!

Hello Melanie,

The moment of order (j,k)   of the variables  X,Y  calculated from the sample of  n  pairs  of results of   n  measurements  (x1,y1),  (x2,y2), . . . , (xn,yn)  is defined, as you definitely know by

m(j,k) =  average of xj yk  = [ (x1)j (y1)k + . . . . + (xn)j (yn)k  ] / n

For   j=1, k=0  it is just the mean of 

x1, x2,...,xn,  

for (j,k) = (1,1), i.e.  j=1 k=1  this is the  mixed moment of  order  1,1  which is the mean value of the products 

x1 y1,  x2 y2,..., xn yn. 

Note, that the sample covariance equals   m(1,1) - m(1,0) m(0,1)  usually denoted  by   cov(x,y) or  sx,y   . 

Let me finish the basic explanations  with the note, that  m(0,2) - ( m(0,1) )2  is the sample variance of variable  Y  denoted in some book by   sY2.  

Next, the additional sub-index  l =1, or l=2 used in my answer,  indicates that the moments should be calculated  for female  or male part separately. Then the sizes are   n1  or  n2  respectively, (146  or 30, respectively for your data). For more detailed definitions please consult any book in statistics. Unfortunately, the RG editor is difficult to manage with mathematical formulas  :(

Best regards, Joachim

PS. I am not familiar with the statistical packages, but within their "Help" everything should be available under  sample moments, cf.also:

http://mathworld.wolfram.com/Covariance.html

http://mathworld.wolfram.com/CorrelationCoefficient.html

http://www.le.ac.uk/users/dsgp1/COURSES/LEISTATS/Lecture6.pdf

http://www.r-tutor.com/elementary-statistics/numerical-measures (in particular the subchapters: mean variance, standard deviation, covariance, correlation coefficient, central moments)

Regards, Joachim

Melanie wrote:

"1) I conducted a survey with 176 people, answering all questions related to the variables. However to do the regression analysis I split the genders into f & m, which were distributed unequally. So I reduced the number of female to match the amount of male participants. So now I did a regression using all participants to find the relation between X and Y and another two to measure the impact of females and males. Females had a higher impact, and now I need to calculate the final impact of X and X2 together on Y. -- I hope this explains my situation a bit better and answers your first question."

I think what you're saying is that you want to know whether nature of the X-Y relationship depends on sex (your X2). If so, and assuming X2 has values 1 and 0, just estimate a model as follows:

Y = b0 + b1X + b2X2 + b3X*X2 + error

The b1 coefficient = the X-Y slope for the group with X2=0, and b3 = the slope for the X2=1 group minus the slope for the X2=0 group. The t-test for b3 tests the null hypothesis that the slopes for the two populations are equal.

Finally, there is no requirement that the numbers of males & females be equal. So I would not discard data.

HTH.

I like the idea by Bruce, which is very reasonable if the aim is to test the hypothesis "b3=0" (or, equivalently, of the equality of the slopes in the two groups). Let me add that then b2 is to be interpreted as a mean shift of the influence of  the gender upon  Y.  Thus, isn't b2  to be added it to the investigation. Obviously only then if the slope is not  the only problem.  This can be important in some other problems. But, of course,  if you need to apply the results to predict  increase  of  Y  as a consequence of an increase  X  within the "total" population,  the averages given  by formulas  (0) (or improved  by something like (0') )  are  seemingly appropriate. Then there is an open problem how one should choose the weights which take into account the different sizes of the groups.

Regards, Joachim