Chi square test of independence and categorical variable

[#1]

I am quite new to statistics. I have designed an experiment where participants are supposed to tap to reproduce the tempo of some music samples. I have manipulated the music samples at tempo values at 60, 120 and 150 bpm, which is my x-variable and I have labelled it as categorical and nominal. In addition to the recorded tempo values (y-variable), I am looking at one more variable which I call the S-factor. The S-factor is a categorical variable and provides me information if the given tempo and the reproduced tempo match. For eg. If 60 bpm was perceived at 60 bpm, the label is 1 = Same. If 60 was perceived at 120 then the label is 2=Double. There are 4 labels in S-factor 0- Half, 1= Same, 2 = Double, 3 = Other. 

My H0 is = The perceived tempo is same as given tempo which means 60 will be perceived at 60, 120 will be perceived at 120 and 150 at 150.

H1 = The perceived tempo is not same as given tempo. 

I have two questions:

1) Is Given tempo (60bpm, 120 bpm and 150 bpm) at which the songs are presented a categorical variable or continuous variable? Should S-factor be labelled as nominal or ordinal variable?

2) In order to test the hypothesis, should I do the chi-square test of independence since both Given Tempo and perceived tempo are categorical or Should I just go to analyse ---> Fit Model ----> Put Tempo as x-variable and S-Factor as y-variable? I do not if these are two different approaches of looking at the problem or only one of them is correct?

[#2 · Accepted solution]

Generally use continuous factors as much as possble. You only presented three levels but there is an infinity of choices for tempo but the factor is still continuous.

S is not a factor. It is another (derived) response. It is redundant with your response variable Y, the recorded tempos. Do you measure the actual tempo or classify it into one of three levels? If it is the former, then use a continuous numerica data column. If it the latter, then use an ordinal data column.

I think you only have two variables so you can use Fit Y by X to begin your analysis. I am not sure if it will launch Logistic or Contingency. It depends on the answer to my question above.

[#3]

I do measure the actual tempo but I call it pulsation (y-variable) since it is the pulsation which is subjective and recorded as response. I have labeblled the pulsation as continuous variable. For each song at a given tempo, I record its corresponding pulsation rate. 

I now understand why the given tempo should also be a continuous variable! Thank you.

Once I get the pulsation, I derive the S-variable and assign labels. I am still unable to understand why S should be ordinal and not nominal. 

[#4]

Your description of S led me to believe that the categorical levels indicated an increasing difference: 1 < 2 < 3. So I would use the ordinal modeling type in such a case.

I recommend that you fit a model of y versus x and then use predicted y to compute S. I think the quality of S will be higher than if you assign S based on observed y and then try to analyze or model it.

[#5]

I guess I did not define S correctly. It does have labels increasing order but they are not really in order of incrementation. 

For eg., which means if a song presented at 60 bpm was perceived at 30 bpm. Then the pulsation will be 30 but S-factor is Half or I have the value label for it is 0. If 60 bpm was perceived at 60 bpm then the S-factor will be Same or value label 1. Each song is independent of each other, so I assume that labels are also independently assigned depending only on given tempo value and pulsation rate.

In such a case is S still ordinal?

[#6]

First, how would you use the S = Other level in your assignment? Is it for any outcome that does not exactly match Half, Same, or Double?

Second, why convert a quantitative variable into a qualitative variable? There is always loss of information. I guess it is for interpretation, not analysis.

Third, do you have the data already or are you still planning the study?

Last, given that the presented tempo (X) and the observed tempo (Y) are both continuous numeric variables, you will use regresion analysis. The analysis of variance will present the F ratio instead of Chi square. A t ratio is presented for each term. Examine the data in a scatter plot first to determine if there are any data problems, if there is a non-random pattern, and the possible complexity necessary to model the data (e.g., linear, quadratic, cubic, et cetera). This can all be done through Fit Y by X > Bivariate platform.

[#7]

Yes, there are outcomes for which S does not match Half, Same, or Double. But they are very few. To elaborate, I have 84 music clips and each of the 28 music clips are set at specific tempo values of 60, 120 and 150 bpm i.e., 28 clips set at 60 bpm, 28 clips set at 120 bpm and 28 clips set at 150 bpm.

If the pulsation rate for given tempo 120 is obtained at 119, I label S as Double for that value. However, if pulsation for 120 is obtained as 80 it is labelled as Other.

I already have the data. I am now running into problems with Statistical analysis.

[#8]

Yes, there are outcomes for which S does not match Half, Same, or Double. But they are very few. To elaborate, I have 84 music clips and each of the 28 music clips are set at specific tempo values of 60, 120 and 150 bpm i.e., 28 clips set at 60 bpm, 28 clips set at 120 bpm and 28 clips set at 150 bpm.

If the pulsation rate for given tempo 120 is obtained at 119, I label S as Double for that value. However, if pulsation for 120 is obtained as 80 it is labelled as Other.

I already have the data. I am now running into problems with Statistical analysis.

[#9]

Yes, there are outcomes for which S does not match Half, Same, or Double. But they are very few. To elaborate, I have 84 music clips and each of the 28 music clips are set at specific tempo values of 60, 120 and 150 bpm i.e., 28 clips set at 60 bpm, 28 clips set at 120 bpm and 28 clips set at 150 bpm. If the pulsation rate for given tempo 120 is obtained at 119, I label S as Double for that value. However, if pulsation for 120 is obtained as 80 it is labelled as Other. I already have the data. I am now running into problems with Statistical analysis.