An example of linearity verification
| Level | PH | Measured value | σ value | Weight | Predicted value (P) | Deviation | 90% CI | ±ADL | Overlaps with ±ADL | |||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| #1 | #2 | #3 | Mean (M) | SD | Lower limit | Upper limit | ||||||||
| 1 | 1 | 3,250 | 3,393 | 3,301 | 3,314.7 | 72.473 | 70.658 | 1/(70.658)2 | 3,234.93 | 79.74 | -17.36 | 176.83 | ±161.75 | Acceptable |
| 2 | 0.75 | 2,502 | 2,401 | 2,378 | 2,427.0 | 65.962 | 51.736 | 1/(51.736)2 | 2,435.03 | -8.03 | -79.12 | 63.06 | ±121.75 | Acceptable |
| 3 | 0.5 | 1,619 | 1,650 | 1,639 | 1,636.0 | 15.716 | 34.874 | 1/(34.874)2 | 1,635.13 | 0.87 | -47.05 | 48.79 | ±81.76 | Acceptable |
| 4 | 0.25 | 788 | 790 | 799 | 792.3 | 5.859 | 16.890 | 1/(16.890)2 | 835.24 | -42.90 | -66.11 | -19.69 | ±41.76 | Acceptable |
| 5 | 0.1 | 331 | 340 | 342 | 337.7 | 5.859 | 7.198 | 1/(7.198)2 | 355.30 | -17.63 | -27.52 | -7.74 | ±17.76 | Acceptable |
| 6 | 0 | 35 | 36 | 35 | 35.3 | 0.577 | 1/(0.577)2 | 35.34 | 0.00 | -0.80 | 0.79 | ±1.77 | Acceptable | |
| A(WLS) | 3,199.6 | B(WLS) | 35.336 | |||||||||||
Construct a function with no intercept: Y=S∙X (X-axis: M; Y-axis: SD)
When calculating the slope S using Microsoft Excel, use the command ‘LINEST((SD region), (M region),0)’ instead of the command ‘slope’ because it presupposes that there is a non-zero intercept.
Then, σ=S∙M
Construct a function of the least square linear regression: Y=A∙X+B (X-axis: PH, Y-axis: M)
When calculating the slope A using Microsoft Excel, use the command ‘LINEST((M region), (PH region)^2)’. And when calculating the intercept B using Microsoft Excel, use the command “SQRT(SUMSQ(Level 6 data)/3), where 3 is the number of replicates in this example.
Then, P=A∙PH+B
Deviation=M–P; 90% CI=Deviation±
Abbreviations: PH, the proportion of the high-level sample; SD, standard deviation; CI, confidence interval; ADL, allowable deviation from linearity; WLS, weighted linear square regression.