Regression sum of squares (SSR) measures the variability in observed responses explained by a fitted regression. For response values y, predictions ŷ, and response mean ȳ, calculate SSR as Σ(ŷ − ȳ)². Unlike total sum of squares, SSR uses predictions rather than observed values and excludes unexplained residual variation. section 4.2.3
Use SSR with the other components of the regression sum-of-squares decomposition:
- SST = Σ(y − ȳ)² measures total response variability without predictors. section 4.2.3
- SSE = Σ(y − ŷ)² measures the portion not explained by the regression. section 4.2.3
- SST = SSR + SSE, and R² = SSR / SST; therefore, SSR indicates the explained share of total variability. section 4.2.3
Calculating SSR with Python
After fitting a linear model named lm to predict stage2 from stage1, calculate the explained variability with `SSR = np.sum((lm.predict(x) - np.mean(mds['stage2'])) ** 2)`. The marathon-stage example returns SSR = 28,537.79, while SST = 49,636.95, producing R² ≈ 0.575. This value indicates that the fitted relationship explains about 57.5% of stage2 variance; model significance and predictor usefulness still require evaluation. section 4.2.3
Table 4.2 The sum of squares summary for the fitted linear regression. The stage1 and stage2 columns represent, respectively, the observed values from the two mds variables. SST equals the sum of squares total, or the total variability in the dependent variable without the consideration of any predictors. The yhat column contains the predictions for the response variable from the fitted regression. SSE equals the sum of squares error, or the portion of the SST that can’t be explained by the regression; and SSR equals the sum of squares regression, or the percentage of the variance in the dependent variable explained by the model.
|
x
stage1
|
y
stage2
|
(
y −
ȳ)
2
SST
|
y
yhat
|
(
y −
ŷ)
2
SSE
|
(
ŷ −
ȳ)
2
SSR
|
|---|---|---|---|---|---|
|
152
|
191
|
3,654.20
|
195.78
|
22.88
|
3098.73
|
|
162
|
184
|
4,549.50
|
208.87
|
618.36
|
1813.33
|
|
165
|
193
|
3,416.40
|
212.79
|
391.71
|
1494.46
|
|
163
|
191
|
3,654.20
|
210.18
|
367.68
|
1,703.62
|
|
166
|
193
|
3,416.40
|
214.10
|
445.21
|
1,395.02
|
|
187
|
292
|
1,644.30
|
241.57
|
2,542.75
|
97.53
|
|
175
|
235
|
270.60
|
225.87
|
83.27
|
654.10
|
|
202
|
226
|
647.70
|
261.20
|
1,238.96
|
95.04
|
|
179
|
214
|
1,402.50
|
231.11
|
292.68
|
413.80
|
|
222
|
274
|
508.50
|
287.36
|
178.62
|
1,289.87
|
|
184
|
300
|
2,357.10
|
237.65
|
3,887.60
|
190.46
|
|
193
|
242
|
89.30
|
249.42
|
55.12
|
4.10
|
|
255
|
279
|
759.00
|
330.54
|
2,656.24
|
6,255.02
|
|
202
|
288
|
1,335.90
|
261.20
|
718.30
|
95.04
|
|
221
|
269
|
308.00
|
286.06
|
290.92
|
1,197.61
|
|
207
|
329
|
6,014.00
|
267.74
|
3,752.75
|
265.37
|
|
227
|
268
|
273.90
|
293.91
|
671.14
|
1,802.54
|
|
239
|
315
|
4,038.60
|
309.61
|
29.10
|
3,382.11
|
|
164
|
201
|
2,545.20
|
211.48
|
109.90
|
1,597.33
|
|
226
|
345
|
8,751.60
|
292.60
|
2,745.97
|
1,693.16
|
|
49,636.95
|
21,099.16
|
28,538.24
|
Inspect the sum-of-squares table to compare observed responses, fitted predictions, residual errors, SST, SSE, and SSR row by row. section 4.2.3
