The short answer is that you interpret quantile regression coefficients just like you do ordinary regression coefficients. The long answer is that you interpret quantile regression coefficients almost just like ordinary regression coefficients.
We can illustrate this with a couple of examples using the hsb2 dataset.
use https://stats.idre.ucla.edu/stat/stata/notes/hsb2, clear
tabstat write, by(female) stat(p25 p50 p75)
Summary for variables: write
by categories of: female
female | p25 p50 p75
-------+------------------------------
male | 41 52 59
female | 50 57 62
-------+------------------------------
Total | 45.5 54 60
--------------------------------------
We will begin by running median and .75 quantile regression models without any predictors.
qreg write
Iteration 1: WLS sum of weighted deviations = 1595.95
Iteration 1: sum of abs. weighted deviations = 1591
Iteration 2: sum of abs. weighted deviations = 1571
Median regression Number of obs = 200
Raw sum of deviations 1571 (about 54)
Min sum of deviations 1571 Pseudo R2 = 0.0000
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons | 54 1.239519 43.57 0.000 51.55572 56.44428
------------------------------------------------------------------------------
qreg write, quantile(.75)
Iteration 1: WLS sum of weighted deviations = 1237.9502
Iteration 1: sum of abs. weighted deviations = 1202.5
Iteration 2: sum of abs. weighted deviations = 1084.5
75 Quantile regression Number of obs = 200
Raw sum of deviations 1084.5 (about 60)
Min sum of deviations 1084.5 Pseudo R2 = 0.0000
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons | 60 .6665574 90.01 0.000 58.68558 61.31442
------------------------------------------------------------------------------
In the median regression the constant is the median of the sample while in the .75 quantile regression the constant is the 75th percentile for the sample.
Next, we’ll add the binary predictor female to the model.
qreg write female
Iteration 1: WLS sum of weighted deviations = 1543.9433
Iteration 1: sum of abs. weighted deviations = 1545
Iteration 2: sum of abs. weighted deviations = 1542
Iteration 3: sum of abs. weighted deviations = 1536
Median regression Number of obs = 200
Raw sum of deviations 1571 (about 54)
Min sum of deviations 1536 Pseudo R2 = 0.0223
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 5 2.611711 1.91 0.057 -.1503394 10.15034
_cons | 52 1.927268 26.98 0.000 48.19939 55.80061
------------------------------------------------------------------------------
predict p50
(option xb assumed; fitted values)
tabulate p50
Fitted |
values | Freq. Percent Cum.
------------+-----------------------------------
52 | 91 45.50 45.50
57 | 109 54.50 100.00
------------+-----------------------------------
Total | 200 100.00
qreg write female, quantile(.75)
Iteration 1: WLS sum of weighted deviations = 1204.3893
Iteration 1: sum of abs. weighted deviations = 1272
Iteration 2: sum of abs. weighted deviations = 1154.5
Iteration 3: sum of abs. weighted deviations = 1060
75 Quantile regression Number of obs = 200
Raw sum of deviations 1084.5 (about 60)
Min sum of deviations 1060 Pseudo R2 = 0.0226
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 3 1.23163 2.44 0.016 .5712035 5.428796
_cons | 59 .9385943 62.86 0.000 57.14908 60.85092
------------------------------------------------------------------------------
predict p75
(option xb assumed; fitted values)
tabulate p75
Fitted |
values | Freq. Percent Cum.
------------+-----------------------------------
59 | 91 45.50 45.50
62 | 109 54.50 100.00
------------+-----------------------------------
Total | 200 100.00
From this point on I’ll describe what is going on in the median regression model. The interpretation for the .75 quantile regression is basically the same except that you substitute the term 75th percentile for the term median.
With the binary predictor, the constant is median for group coded zero (males) and the coefficient is the difference in medians between males and female (see the tabstat above).
Looking at the tabulated predicted scores we see that we get two values, the conditional median for males (52) and the conditional median for female (57).
Now, let me show you something that is really neat about quantile regression. I will replace the highest value of write (67) with the value of 670 and rerun these analyses.
replace write=670 if write==67
(7 real changes made)
qreg write female
Iteration 1: WLS sum of weighted deviations = 8319.5083
Iteration 1: sum of abs. weighted deviations = 6544
Iteration 2: sum of abs. weighted deviations = 6156
Iteration 3: sum of abs. weighted deviations = 5757
Median regression Number of obs = 200
Raw sum of deviations 5792 (about 54)
Min sum of deviations 5757 Pseudo R2 = 0.0060
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 5 2.611711 1.91 0.057 -.1503394 10.15034
_cons | 52 1.927268 26.98 0.000 48.19939 55.80061
------------------------------------------------------------------------------
qreg write female, quantile(.75)
Iteration 1: WLS sum of weighted deviations = 11445.07
Iteration 1: sum of abs. weighted deviations = 7582
Iteration 2: sum of abs. weighted deviations = 7461
Iteration 3: sum of abs. weighted deviations = 7391.5
75 Quantile regression Number of obs = 200
Raw sum of deviations 7416 (about 60)
Min sum of deviations 7391.5 Pseudo R2 = 0.0033
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
female | 3 1.23163 2.44 0.016 .5712035 5.428796
_cons | 59 .9385943 62.86 0.000 57.14908 60.85092
------------------------------------------------------------------------------
Notice that neither the coefficients nor the standard errors changed. This is because changing this extreme score does not change either the median or the 75th percentile. The only changes that affect the results are when a value crosses a quantile boundary. For example, changing a value of 58 to 580 would not affect the median but would affect the 75th percentile.
For the last example, we will reload the data and use a continuous predictor in the model.
use https://stats.idre.ucla.edu/stat/stata/notes/hsb2, clear
qreg write socst
Iteration 1: WLS sum of weighted deviations = 1219.9071
Iteration 1: sum of abs. weighted deviations = 1219.9333
Iteration 2: sum of abs. weighted deviations = 1212.8
Iteration 3: sum of abs. weighted deviations = 1212.5667
Iteration 4: sum of abs. weighted deviations = 1209.375
Iteration 5: sum of abs. weighted deviations = 1208.9
Median regression Number of obs = 200
Raw sum of deviations 1571 (about 54)
Min sum of deviations 1208.9 Pseudo R2 = 0.2305
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
socst | .6333333 .0571053 11.09 0.000 .5207206 .7459461
_cons | 20.03333 3.069487 6.53 0.000 13.98025 26.08642
------------------------------------------------------------------------------
predict double p50
(option xb assumed; fitted values)
qreg write socst, quantile(.75)
Iteration 1: WLS sum of weighted deviations = 992.87
Iteration 1: sum of abs. weighted deviations = 1003.2667
Iteration 2: sum of abs. weighted deviations = 950.85
Iteration 3: sum of abs. weighted deviations = 936.30001
Iteration 4: sum of abs. weighted deviations = 928.66667
Iteration 5: sum of abs. weighted deviations = 926.07501
Iteration 6: sum of abs. weighted deviations = 924.30001
75 Quantile regression Number of obs = 200
Raw sum of deviations 1084.5 (about 60)
Min sum of deviations 924.3 Pseudo R2 = 0.1477
------------------------------------------------------------------------------
write | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
socst | .4 .0408158 9.80 0.000 .3195104 .4804896
_cons | 37.6 2.187081 17.19 0.000 33.28704 41.91296
------------------------------------------------------------------------------
predict double p75
(option xb assumed; fitted values)
With the continuous predictor socst the constant is the predicted value when socst is zero. The quantile regression coefficient tells us that for every one unit change in socst that the predicted value of write will increase by .6333333.
We can show this by listing the predictor with the associated predicted values for two adjacent values. Notice that for the one unit change from 41 to 42 in socst the predicted value increases by .633333.
sort socst
list socst p50 p75 in 42/43
+--------------------------+
| socst p50 p75 |
|--------------------------|
42. | 41 46 54 |
43. | 42 46.633333 54.4 |
+--------------------------+