This document summarizes an analysis of factors that contribute to a primary care physician diagnosing depression in patients. Six explanatory variables were explored in relation to a binary response variable indicating depression diagnosis. Statistical tests showed significant differences in mean vectors between those diagnosed/not diagnosed with depression and education levels. Multivariate regression identified physical and mental health scores and gender as predictive, but not age. Principal component and linear discriminant analyses provided limited differentiation and classification between diagnosis groups due to high variability in predictor variables. While multivariate techniques were useful, clear separation between diagnosis groups was challenging with this data.

1. Introduction
Depression, long stigmatized as a complaint of the merely idle or a “case of the blues,” is
increasingly accepted in the medical community as a serious and possibly fatal disease if left
untreated, but one that can be managed with psychiatric care. For this reason, as well as for the
general preservation of the happiness of the patient, it is worthwhile to investigate what factors
contribute to a primary care physician diagnosing depression in a patient during a visit. Such an
investigation could have all sorts of benefits, such as exposing bias on the part of physicians to
under- or over-diagnose depression in particular demographic groups, the level of trust
physicians put in various metrics (such as the Beck depression score and the SF-36 self-
administered test of mental and physical wellness, both of which are included in the dataset I will
be using for this paper). The data is of particular importance as depression in primary care can
lead to complications in treatment down the road.
Material and Methods
This paper will seek to explore the relationships between six explanatory variables (PCS and MCS,
the physical and mental components of the SF-36 tests measuring physical and mental health of
the patient; BECK, the Beck depression score; PGEND, the patient’s gender; AGE, the patient’s
age; and EDUCAT, the number of years of formal schooling) and the binary response variable
DAV, where 1 represents a diagnosis of depression. A handful of multivariate techniques
immediately suggest themselves when examining the structure of the data, particularly due to
the heterogeneity of types of data in the set (DAV, EDUCAT and PGEND are factors, while PCS,
MCS, AGE, and BECK are continuous numerical variables). Methods used were Hotelling’s T2 and
Wilks’ Lambda tests of the equality of mean vectors, multivariate linear regression, principal
component analysis, and a linear discriminant analysis.
Results
Equality of Means
Using Hotelling’s T2 test, I established that at a significance level α=0.95, there was significant
evidence (P-value = 1.727e-10) to reject equality of mean vectors between those who had been
diagnosed with depression (m1 = 64) and those who had not (m2=336). Using Wilks’ Lambda test,
I was further able to establish inequality of mean vectors (p-value = 2.557e-06) across the three
education groups (years educated, with factor levels Less than 12 years (n1=61), 12 to 14 years
(n2=241), and More than 14 years (n3=90) ). Mean vectors and comparison of results are given
below.
group: no | group: yes | Hotelling's T^2
vars n mean sd median | vars n mean sd median | --------------------
pcs 1 336 41.54 11.44 41.85 | pcs 1 64 38.77 12.04 38.46 | Test stat: 13.725
mcs 2 336 46.41 11.92 48.81 | mcs 2 64 35.00 13.31 34.63 | Numerator df: 4
beck 3 336 4.66 5.04 3.00 | beck 3 64 9.36 7.15 7.00 | Denominator df: 395
age 4 336 41.89 15.74 38.00 | age 4 64 43.66 13.15 42.00 | P-value: 1.727e-10
---------------------------------------------------------------------------------------
group: Less than 12 years | group: 12-14 years | group: More than 14yr |
vars n mean sd median | n mean sd median | n mean sd median |
pcs 1 61 39.30 11.00 37.40 | 249 40.60 11.39 40.75 | 90 43.67 12.15 46.42 |
mcs 2 61 42.38 13.33 42.74 | 249 44.79 12.48 46.91 | 90 45.50 13.45 48.30 |
beck 3 61 7.57 7.59 5.00 | 249 5.13 5.23 4.00 | 90 4.73 5.10 3.50 |
age 4 61 50.77 15.48 53.00 | 249 40.58 15.29 36.00 | 90 40.76 13.55 39.00 |
---------------------------------------------------------------------------------------

2. One-way MANOVA (Bartlett Chi2)
Wilks' Lambda = 0.90261, Chi2-Value = 40.525, DF = 8.000, p-value = 2.557e-06
sample estimates:
pcs mcs beck age
12-14 40.60370 44.79197 5.128514 40.58233
Less than 12 39.29652 42.37531 7.573770 50.77049
More than 14 43.67233 45.49755 4.733333 40.75556
Multivariate Regression
Using R, I fit a multivariate regression model to 𝒀𝒀� = 𝑿𝑿𝜷𝜷� + 𝜺𝜺 such that
𝒀𝒀 = �
𝑌𝑌1,1 𝑌𝑌1,2
⋮ ⋮
𝑌𝑌400,1 𝑌𝑌400,2
� , 𝑿𝑿 = �
1 𝑋𝑋1,1 𝑋𝑋1,2 𝑋𝑋1,3
⋮ ⋮
1 𝑋𝑋400,1
⋮ ⋮
𝑋𝑋400,2 𝑋𝑋400,3
� , 𝜷𝜷� =
⎣
⎢
⎢
⎡
𝛽𝛽0,1
𝛽𝛽1,1
𝛽𝛽0,2
𝛽𝛽1,2
𝛽𝛽2,1
𝛽𝛽3,1
𝛽𝛽2,2
𝛽𝛽3,2⎦
⎥
⎥
⎤
, 𝜺𝜺 ~ 𝑁𝑁(𝟎𝟎, 𝚺𝚺)
Where Yi1 and Yi2 are MCS and BECK respectively; Xi1 , Xi2 , and Xi3 are the values of PCS, AGE,
and PGEND at a given i, 𝛽𝛽̂𝑖𝑖𝑖𝑖 are the values of the parameter estimates, and 𝜺𝜺 is a (400 x 2) noise
matrix. After carrying out Type I and Type II MANOVA tests, however, it would appear that AGE
is not doing much of the heavy predictive lifting:
Df Hotelling-Lawley approx F num Df den Df Pr(>F)
pcs 1 0.088653 17.5089 2 395 5.18e-08 ***
age 1 0.003401 0.6717 2 395 0.5113978
pgend 1 0.036114 7.1325 2 395 0.0009057 ***
Residuals 396
Type II MANOVA Tests: Pillai test statistic
Df test stat approx F num Df den Df Pr(>F)
pcs 1 0.079172 16.9808 2 395 8.419e-08 ***
age 1 0.002663 0.5274 2 395 0.5905750
pgend 1 0.034855 7.1325 2 395 0.0009057 ***
Removal of AGE from the model has very little effect on RSE, R2 or parameter estimates:
mcs beck
(Intercept) 40.256259297 11.55878014
pcs 0.055235813 -0.11995336
age 0.002945113 -0.01477639
pgend1 4.953358482 -1.52212099
mcs beck
(Intercept) 40.40822220 10.7963433
pcs 0.05446585 -0.1160902
pgend1 4.96332993 -1.5721503
Given that this is the case, I felt it reasonable to omit AGE from the final model:
𝒀𝒀� = �
1 𝑥𝑥1,1 𝑥𝑥1,2
⋮ ⋮ ⋮
1 𝑥𝑥400,1 𝑥𝑥400,2
� �
40.41 10.80
0.05 −0.12
4.96 −1.57
� + 𝜺𝜺

3. Principal Component Analysis
A principal component analysis was mildly successful; after standardizing the three variables PCS,
MCS and BECK to be centered at 0 with a standard deviation of 1, principal component analysis
was able to explain ~90% of the variation between samples using two linear combinations P1 and
P2; this analysis was, however, not particularly useful for classifying diagnosis or failure to
diagnose in primary care.
A biplot of the data projected on P1 and P2 confirms that there is not much differentiation
between higher or lower values of each component depending upon the group to which a subject
belongs. The loadings and cumulative contribution to variance for P1, P2 and P3 are given in the
output below.
Rotation: PC1 PC2 PC3
pcs -0.2994318 -0.93485239 -0.1907659
mcs -0.6552573 0.34681932 -0.6710844
beck 0.6935261 -0.07594322 -0.7164176
[sd] 1.3199826 0.9761081 0.5521403
Importance of components:
PC1 PC2 PC3
Standard deviation 1.3200 0.9761 0.5521
Proportion of Variance 0.5808 0.3176 0.1016
Cumulative Proportion 0.5808 0.8984 1.0000

4. Linear Discriminant Analysis
Results from LDA were similarly
underwhelming; as is evident from the
histogram and the scores/classes plots,
not very much separation between
factor levels could be achieved. The
highest density for the group not
diagnosed with depression is clearly
centered around LDA = -1, but there is
no clear highest density for the group
diagnosed, and it meanders into the
LDA = -1 region as well.

5. Values for the LDA coefficients are given below.
lda(dav ~ pcs + mcs + beck + age, data = f.lda)
Group means:
pcs mcs beck age
no 41.53721 46.40679 4.660714 41.89286
yes 38.77215 35.00299 9.359375 43.65625
Coefficients of linear discriminants:
LD1
pcs -0.007293228
mcs -0.054224639
beck 0.076986186
age 0.008000076
IV. Discussion
The tools of multivariate analysis are quite powerful, though it is clear from some of the results
that the utility of these techniques is largely dependent upon the nature of the data being
analyzed and the response variable being modeled. Clear differences in mean vectors were
demonstrated for two different classification groups (diagnosis and level of education), but the
high variability of the continuous predictor variables in the data made PCA and LDA less useful
than I might have liked. Still, some separation was achieved, and it was a powerful illustration
of how useful these techniques can be.