# Case vs Group of Controls

#### STOUCHA

##### New Member
Hi. I am a medical resident doing a required research project. Perhaps someone can help me by pointing me towards the correct statistical test. I am trying to test whether bone density (a numerical value typically between 100 and 400) in a group of patients with a specific disease is different than the general population. I have only a handful of patients with the disease (11 patients) and have measured their bone density. These 11 patients have ages from 14 to 65 and are both male and female. Since bone density is affected by both age and gender I have measured the bone density in 10 control patients matched for age and gender per patient with disease.

So, how do I tell if the bone density of these diseased patients is different than their respective control groups?

Any help would be greatly appreciated. Thanks

#### STOUCHA

##### New Member
Data looks something like this

Patient 1 (male age 25 with disease x) - bone density = 306
Controls (10 males age 25 without disease x) - bone densities = 235, 264, 248, 287, 255, 292,245,226,254,275

Patient 2 (female with disease x, age 60) - bone density = 200
Controls (10 females age 60 without disease x) - bone densities = 150, 135, 165, 186, 134, 155, 165, 143, 176, 123

etc etc for 11 patients.

How do I determine if there is a significant difference in bone density in the diseased patients compared to their controls?

Thanks

Last edited:

#### STOUCHA

##### New Member
Possible Solution.

After some thought, here is my solution. Any feedback would be greatly appreciated.

The null is that the bone densities of the diseased patients is no different than age/sex matched controls. This would mean that both the mean and variance are the same as controls.

Thus, if the variance of the diseased patients is different than the variance of controls then I must reject the null.

So, I do a modified Levene's test as follows:

Disease Patient 1: variance^2 = (bone density - mean)^2 where the mean is the mean of the ten age/sex matched controls for patient 1.

Disease Patient 2: variance^2 = (bone density - mean)^2 where the mean is the mean of the ten age/sex matched controls for patient 2, i.e. different than the mean used for the calculation of patient 1

Disease Patient 3: etc etc

This gives me a list of var^2 for diseased patients, i.e. 11 patients in my study.

Next I calculate the variance for all control patients as follows:

Control Pt 1 from control group for disease patient 1: var^2 = (bone density - mean) ^2 where the mean is the mean of all ten control patients for disease patient 1, i.e. this is the same mean used to calculate the var^1 for disease patient 1 above.

repeat this for all controls for disease patient 1 using the mean of the group of controls for disease patient 1.

Perform this same analysis on the 10 control patients for disease patient 2 now using the mean from this group which is the same mean used to calculate the var^2 of patient 2.

Continue calculating the var^2 for all 110 control patients per this method.

Perform a t-test type analysis on these two groups of var^2.

If it is different then I reject the null.

If I reject the null I would say the meaning is that the bone density in the diseased patients is in some way not similar to the bone density in the controls patients.

Any thoughts, and especially any criticisms, are greatly appreciated.

Last edited:

#### jamesmartinn

##### Member
You're design is a bit confusing to follow. Are you looking to compare patients vs controls irrespective of their specific disease? If so, you can do that simply with a t-test for independent groups. Most software programs will calculate and re-adjust for the unequal sample sizes, including Levene's Test of Homogeneity of Variance.

I know that you've elaborated, but if you could any further about your intentions for this experiment, it would be great.

#### Mean Joe

##### TS Contributor
Maybe you could do two regression lines, using age and gender as predictor variables, and fitting bone density of the controls (one line) and the cases (second line)? There is an assumption of normality in bone densities, as well as some other assumptions.

If the two lines have the same slope, then the vertical distance between the two lines represents the difference in means of the bone densities in the two groups adjusted for any difference in the distribution of the predictors.

This is known as analysis of covariance (ANCOVA).

#### Mean Joe

##### TS Contributor
You're design is a bit confusing to follow. Are you looking to compare patients vs controls irrespective of their specific disease? If so, you can do that simply with a t-test for independent groups. Most software programs will calculate and re-adjust for the unequal sample sizes, including Levene's Test of Homogeneity of Variance.

I know that you've elaborated, but if you could any further about your intentions for this experiment, it would be great.
STOUCHA mentioned that bone density is known to depend on age and sex. Would a t-test, comparing mean of cases vs mean of controls, still be possible?

#### jamesmartinn

##### Member
I suppose you could covary gender and age. Just make sure to test the relevant assumptions of ANCOVA.

also, thanks MeanJoe for looking into this

#### STOUCHA

##### New Member
Thanks for feedback!

Sure. Let me explain a little better. First the actual data:

Patient 1 (age 51 bone density 112)
Controls for patient 1
age mean
51 156
50 99.3
51 180
50 159
51 103
50 149
50 149
51 101
50 119
50 128.7

Patient 2 (age 58 bone density 181)
Controls for patient 2
age mean
58 169
59 128
59 153
57 119
58 120
58 147
58 128
59 163
56 132
57 149

Patient 3 (age 59 bone density 153)
controls for patient 3
59 112
58 133
59 71
59 91
57 114
58 117
58 175
59 101
58 166
58 110

Patient 4: age 39, bone density 151
Controls for patient 4
39 168
40 236
38 210
39 181
40 179
39 161
38 173
39 151
39 154
39 219

Patient 5: age 14, bone density 313
Controls for Patient 5
14 243
14 220
14 241
14 271
13 195
14 228
14 273
13 305
14 230
14 254

Patient 6: age 23, mean 272
Controls for patient 6
22 151
22 195
22 227
23 181
22 179
22 213
22 185
23 253
22 217
23 231

Patient 7: age 22, bone density 238
controls for patient 7
21 166
22 195
21 185
21 228
22 169
22 199
21 270
21 169
21 169
21 176

Patient 8: age 40, bone density 124
controls for patient 8
39 216
39 160
40 187
40 144
39 182
40 155
40 172
40 228
40 180
40 236

Patient 9: age 42, bone density 124
Controls for patient 9
41 176
42 185
41 201
42 186
42 201
41 224
41 193
42 183
44 205
40 176

Patient 10: age 43, bone density 200
Controls for patient 10
42 191
42 163
42 170
42 121
42 167
42 109
43 185
42 206
42 129
42 162

Patient 11: age 44, bone density 151
Controls for patient 11
44 139
43 158
43 98
43 167
43 150
44 217
43 140
43 191
43 200
44 170

#### STOUCHA

##### New Member
Graph.

I tried to put up the graph of the data but i can't seem to. The graph that I have is average bone density of each control group versus age. This shows a general trend towards lower bone density with age...not unsuspected. I have also included the standard deviation of each group as "error bars" at each point. A trend line is added (y = -0.0029x3 + 0.3215x2 - 13.002x + 367.28; R2 = 0.881). Finally, I have plotted the bone density of each cocci patient vs age. What I see is that bone densities of most of the cocci patients are greater than 1 standard deviation from the point estimate of the mean bone density for that age. Some are higher and some are lower than their respective control means. I would expect this from the biology. The question is, is this significant? Do the cocci patients really come from a bone density population that has a variance larger than the variance of the controls (adjusted for age)? Or is it just chance that so many of the cocci patient's bone density fall so far from the control means.

The trick in doing the variance calculation is that there is so much variation with age. I have to figure out how to eliminate that in the calculation. It is sort of a paired Variance test.

It seems that I could normalize the data first and then do a standard F-test or Levene's test but I am not sure exactly how to do that. Or, I could use the standard formula for calculating the Levene's test (or F-test) but use the mean of the control group of for each patient...a sort of Paired Levene's.

Does this make sense.

If someone can tell me how to put my graph up I will do it.

Thanks again!!

Last edited:

#### STOUCHA

##### New Member
I figured out how to show the graphs.

Let me know if you have any problems.

There are two graphs: one is bone density and the other is fat density collected in the same manner as the bone density measurements from the same patients/control patients to serve as a control.

Last edited: