MHB Help to write a set of equations

[kirti]

i am doing a research based project and need help to write a set of equations with variables at two levels.i am clear on what i need to do but don`t know how to express it in equation form.
I have a dependent variable which is a function of 12 parameters.
each parameter has several sub parameters which are basically questions asked from respondents on certain issues at two different time periods.
I have taken the percentage change in the responses for the sub parameter. I have then put a weightage to the percentage change so that I do not have a very wide peaks in data.
so i have assigned points to this change in response as per below -

points %age change[TABLE="width: 927, align: center"]
[/TABLE]

0 0 or less than 0
1 0-25%
2 26%-50%
3 51%-75%
4 76%-100%
5 more than 100%
when i add up the points, i hope to get a sum of how each subparameter is influencing the parameter. further, by putting how many different subparameters come under different parameters, i can get how much each parameter is affecting the dependent variable.
i need to write two sets of equations. one for the parameters and then one for the subparameters and i need help.

 

[Ackbach]

I think labels would be of immense help here. Let's call your dependent variable $y$. You have twelve parameters, and they all appear to be of the same kind of variable. So let's use an indexing scheme to represent those: $x_{j},\;j=1,2,\dots,12$. That is, your twelve parameters are $x_{1},x_{2},\dots,x_{12}$. Then you have these subparameters which determine the parameters. One of your subparameters is percent change. Before I can come up with a decent scheme for labeling the subparameters, I have some questions about the parameters and subparameters.

1. Is another subparameter the weight you're assigning?
2. How many subparameters are there?
3. How do the parameters depend on the subparameters?
4. How does the dependent variable depend on the parameters? For example, are you just summing the parameters to get the dependent variable?

We'll go from here, I think.

 

[kirti]

Thanks for the response.
let me clarify. my subparameters are actually questions defining the parameter chosen. like if parameter is human resource decisions, the sub parameters are questions related to this area.
what i have done is recorded the responses to these subparameters at two different time frame(two different years). the percentage change in the responses to the questions will give me some numbers but since the parameters are different variables,just adding the percentage change disnot make sense to me . hence , I thought of putting weightage to the percentage change as given below.
So hope it is a bit clearer.I have also answered point by point to yr. queries below.
Thanks again for the prompt reply.
................................

I think labels would be of immense help here. Let's call your dependent variable $y$. You have twelve parameters, and they all appear to be of the same kind of variable. So let's use an indexing scheme to represent those: $x_{j},\;j=1,2,\dots,12$. That is, your twelve parameters are $x_{1},x_{2},\dots,x_{12}$. Then you have these subparameters which determine the parameters. One of your subparameters is percent change. Before I can come up with a decent scheme for labeling the subparameters, I have some questions about the parameters and subparameters.

1. Is another subparameter the weight you're assigning? - No, explained above
2. How many subparameters are there? -Since they are different questions for each parameter so number of subparameters differs for each parameter.
3. How do the parameters depend on the subparameters?- sub parameters describe the parameters, sub parameters give quantitative value by way of resonses.
4. How does the dependent variable depend on the parameters? For example, are you just summing the parameters to get the dependent variable?- Yes. by summing up the parameters is my idea.

We'll go from here, I think.

 

[Ackbach]

Ok, thanks for the clarifications. I think we're getting closer. Let me see if I've got this right: you've got subparameters that are percent changes to questions asked at different times. You are weighting each of these subparameters in a sum to get each parameter, and then summing those, in turn, to get the value of the independent variable. Is that correct? If so, let me see if I can break out the expressions further.

You've already said that you're summing the parameters to get the independent variable. So we write
$$y=\sum_{j=1}^{12}x_{j}=x_{1}+x_{2}+\dots+x_{12}.$$
So there's that equation done. Now we need to get each $x_{j}$ written down. For each $x_{j}$, we'll need $a_{j}$ subparameters. So $a_{1}$ might be 3 (questions), and $a_{2}$ might be 5 (questions). The $a_{j}$'s are the number of questions making up parameter $x_{j}$. Ok? Now for the questions themselves, we're going to need double-indexing in order to reference the parameter as well as the particular question. I'm going to do $q_{j,k}$, with apologies to the physicists working in relativity, for the answer to the $k$th question pertaining to parameter $j$ in the first year, and $p_{j,k}$ for the answer to the $k$th question pertaining to parameter $j$ in the second year. The percent change for a particular question would then be written as
$$\frac{|q_{j,k}-p_{j,k}|}{|q_{j,k}|}\times 100\%.$$
We also need a weight for this percent change: it also must reference the parameter and the question, so it needs double-indexing. I'll call it $w_{j,k}$. So it seems to me that if you're just going to sum the percent changes weighted by whatever weights you're going to use, then you'd write
$$x_{j}=\sum_{k=1}^{a_{j}}\left[w_{j,k}\cdot\frac{|q_{j,k}-p_{j,k}|}{|q_{j,k}|}\times 100\%\right].$$
Finally, you could combine the two equations I've come up with here into one:
$$y=\sum_{j=1}^{12}\sum_{k=1}^{a_{j}}\left[w_{j,k}\cdot\frac{|q_{j,k}-p_{j,k}|}{|q_{j,k}|}\times 100\%\right].$$

How close are we to what you need?

 

[kirti]

Thanks a ton!This seems like what i wanted.I will put in my values and see the results. Again thank you for your time and effort.
................................

Ok, thanks for the clarifications. I think we're getting closer. Let me see if I've got this right: you've got subparameters that are percent changes to questions asked at different times. You are weighting each of these subparameters in a sum to get each parameter, and then summing those, in turn, to get the value of the independent variable. Is that correct? If so, let me see if I can break out the expressions further.

You've already said that you're summing the parameters to get the independent variable. So we write
$$y=\sum_{j=1}^{12}x_{j}=x_{1}+x_{2}+\dots+x_{12}.$$
So there's that equation done. Now we need to get each $x_{j}$ written down. For each $x_{j}$, we'll need $a_{j}$ subparameters. So $a_{1}$ might be 3 (questions), and $a_{2}$ might be 5 (questions). The $a_{j}$'s are the number of questions making up parameter $x_{j}$. Ok? Now for the questions themselves, we're going to need double-indexing in order to reference the parameter as well as the particular question. I'm going to do $q_{j,k}$, with apologies to the physicists working in relativity, for the answer to the $k$th question pertaining to parameter $j$ in the first year, and $p_{j,k}$ for the answer to the $k$th question pertaining to parameter $j$ in the second year. The percent change for a particular question would then be written as
$$\frac{|q_{j,k}-p_{j,k}|}{|q_{j,k}|}\times 100\%.$$
We also need a weight for this percent change: it also must reference the parameter and the question, so it needs double-indexing. I'll call it $w_{j,k}$. So it seems to me that if you're just going to sum the percent changes weighted by whatever weights you're going to use, then you'd write
$$x_{j}=\sum_{k=1}^{a_{j}}\left[w_{j,k}\cdot\frac{|q_{j,k}-p_{j,k}|}{|q_{j,k}|}\times 100\%\right].$$
Finally, you could combine the two equations I've come up with here into one:
$$y=\sum_{j=1}^{12}\sum_{k=1}^{a_{j}}\left[w_{j,k}\cdot\frac{|q_{j,k}-p_{j,k}|}{|q_{j,k}|}\times 100\%\right].$$

How close are we to what you need?

 

[Ackbach]

You're very welcome! Glad I could be of service.