14  Linearity Analysis

Linearity evaluates whether measured results have an acceptable linear relationship with analyte concentration or target value across the working range. This chapter uses the dedicated linearity() workflow to summarize levels, apply weights, calculate straight-line residuals, compare allowable deviation from linearity (ADL), and perform exploratory polynomial analysis.

ImportantScope of Use

The core of CLSI EP06, 2nd edition, is comparison of straight-line residuals with a prespecified ADL. A high correlation or a nonsignificant quadratic term alone does not establish linearity. method="polynomial" reproduces the superseded EP6-A approach and should be labeled exploratory. The functions do not replace professional judgment about sample preparation, commutability, endpoint coverage, interference, or the reporting range.

flowchart LR
    A["Design LOW/HIGH and dilution levels"] --> B["linearity_replicates()<br/>linearity_panel()"]
    B --> C["Replicate long table"]
    C --> D["linearity()<br/>Aggregate and fit"]
    D --> E["Residuals, intervals, ADL"]
    E --> F["linearity_endpoints()<br/>Check endpoints"]
    F --> G["Claim linearity only within<br/>the validated range"]

Code 
library(ivdtools)
library(readr)

14.1 Main Functions

Function Main purpose Key input or output
linearity_replicates() Calculate replicates per level from precision and ADL imprecision, adl, family_error
linearity_panel() Design a HIGH/LOW preparation series proportion_high, high, low, total_volume
linearity_endpoints() Calculate laboratory-confirmed LOW/HIGH targets LLoQ/ULoQ and low/high-end CV
linearity() EP06 parameterized linearity analysis method, weights, conf.level, adl_*
plot() Plot fit, residuals, precision, or model differences type
Code 
linearity_replicates(imprecision = 4, adl = 5, levels = 8, family_error = 0.10)
  Type: relative
    Imprecision           : 4
    ADL                   : 5
    True deviation        : 0
    Effective ADL         : 5
    Level probability     : 0.98691628
    Z                     : 2.4814824
    Calculated replicates : 3.940963
    Replicates            : 4
    Recommend four        : FALSE
Code 
linearity_panel(proportion_high = seq(0, 1, length.out = 6), high = 1000, low = 10, total_volume = 1)
  Sample: S1
    Proportion high        : 0
    Proportion low         : 1
    Expected               : 10
    Relative concentration : 0.01
    Total volume           : 1
    High volume            : 0
    Low volume             : 1

  Sample: S2
    Proportion high        : 0.2
    Proportion low         : 0.8
    Expected               : 208
    Relative concentration : 0.208
    Total volume           : 1
    High volume            : 0.2
    Low volume             : 0.8

  Sample: S3
    Proportion high        : 0.4
    Proportion low         : 0.6
    Expected               : 406
    Relative concentration : 0.406
    Total volume           : 1
    High volume            : 0.4
    Low volume             : 0.6

  Sample: S4
    Proportion high        : 0.6
    Proportion low         : 0.4
    Expected               : 604
    Relative concentration : 0.604
    Total volume           : 1
    High volume            : 0.6
    Low volume             : 0.4

  Sample: S5
    Proportion high        : 0.8
    Proportion low         : 0.2
    Expected               : 802
    Relative concentration : 0.802
    Total volume           : 1
    High volume            : 0.8
    Low volume             : 0.2

  Sample: S6
    Proportion high        : 1
    Proportion low         : 0
    Expected               : 1000
    Relative concentration : 1
    Total volume           : 1
    High volume            : 1
    Low volume             : 0
Code 
linearity_endpoints(lloq = 10, uloq = 1000, cv_high = 5, cv_at_lloq = 20, cv_at_k_lloq = 15)
CLSI EP06-Ed2 target concentrations
  LOW target: 14.14116
  HIGH target: 924.0072

The replicate-count function reports a design quantity only. Final level and replicate counts must satisfy the protocol, applicable standard, and available samples. LOW/HIGH calculations use laboratory-confirmed target concentrations and are not a substitute for a manufacturer’s validation process.

14.2 Model, Weights, and ADL

Code 
linearity_replicates(
  imprecision, adl, type = c("relative", "absolute"),
  level_probability = 0.99, levels = NULL, family_error = NULL,
  true_deviation = 0, min_replicates = 2
)
linearity_panel(proportion_high, high, low = 0, total_volume = NULL, sample = NULL)
linearity(
  data, sample, result, x, method = c("linear", "polynomial"),
  intercept = TRUE,
  weights = c("equal", "level_variance", "pooled", "precision_profile"),
  variance_group = NULL, conf.level = NULL,
  multiplicity = c("familywise", "none"), adl = NULL,
  adl_abs = NULL, adl_rel = NULL
)

If the per-level central coverage is \(P_L\), repeatability is \(s\), expected true deviation is \(δ_0\), and effective ADL is \(A_{\mathrm{eff}}=A-δ_0\), the replicate count is

\[n=\left\lceil\left(\frac{z_{(1+P_L)/2}s}{A_{\mathrm{eff}}}\right)^2\right\rceil.\]

For familywise error \(α_F\) over \(L\) levels, \(P_L=(1-α_F)^{1/L}\). Relative inputs use percentage units: 5 means 5%.

With high-sample proportion \(p_h\),

\[x_{\mathrm{expected}}=p_hx_H+(1-p_h)x_L.\]

linearity(method="linear") first summarizes each level as \(\bar y_j,s_j,n_j\) and fits \(\bar y_j=\beta_0+\beta_1x_j+e_j\).

weights Level-mean weight Meaning
equal \(1\) Equal weight
level_variance \(n_j/s_j^2\) Use the variance at each level
pooled \(n_j/\widehat\sigma_g^2\) Use groups supplied in variance_group; no automatic grouping
precision_profile \(1/\widehat\sigma_j^2\) Use variance predicted from an SD–mean profile

The residual is \(r_j=\bar y_j-\widehat y_j\), and ADL is exceeded when \(|r_j|>ADL_j\). When weights="pooled", groups must be explicit, contiguous after sorting by \(x\), and contain at least two levels. percent_base determines the denominator of relative residuals. multiplicity="familywise" converts overall coverage to per-level coverage; "none" uses the overall coverage at every level.

14.3 Analysis Examples

14.3.1 Serial-Dilution Linearity

Code 
dilution <- read_csv("./data/014-线性分析/linearity-dilution.csv", show_col_types = FALSE)
dilution <- as.data.frame(dilution)
head(dilution, 6)
  conc replicate signal
1   50         1   48.6
2   50         2   49.2
3   50         3   50.0
4  100         1  101.3
5  100         2   99.7
6  100         3  101.4
Code 
fit_lin <- linearity(
  dilution, sample = "conc", result = "signal", x = "conc",
  method = "linear", weights = "equal", conf.level = 0.95,
  multiplicity = "familywise", adl_rel = 5
)
print(fit_lin)
Linearity regression analysis

Analysis settings

Method: linear (EP06-Ed2)
Observations / levels: 24 / 8
Intercept: estimated
Weights: equal
Fit unit: level_mean

Model comparison

Degree: 1
Observations                : 8
Parameters                  : 2
Residual degrees of freedom : 6
RSS                         : 146.65863
Weighted RSS                : 146.65863
SY.X                        : 4.9439969
Weighted SY.X               : 4.9439969
R-squared                   : 0.99999571
Adjusted R-squared          : 0.999995

Regression coefficients

     Estimate  Std. Error    t value Pr(>|t|)    
a 0.975290473 2.205878722    0.44213  0.67388    
b 0.998893935 0.000844269 1183.14720  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Sample fit information

Sample  Expected  Predicted  Observed mean  Weight  Observed SD  n
50            50     50.920        49.2667       1     0.702377  3
100          100    100.865       100.8000       1     0.953939  3
200          200    200.754       202.6667       1     4.452340  3
400          400    400.533       397.8333       1     7.396170  3
800          800    800.090       798.8667       1    10.261741  3
1600        1600   1599.206      1596.9000       1    13.753545  3
3200        3200   3197.436      3207.8000       1    20.940869  3
6400        6400   6393.896      6389.5667       1    19.923437  3

Sample bias information

Sample   Residual  Recovery bias  Residual (%)  Recovery bias (%)        ADL  Decision  
50      -1.653321      -0.733333    -3.2468990          -1.466667    2.54600  Within ADL
100     -0.064684       0.800000    -0.0641295           0.800000    5.04323  Within ADL
200      1.912589       2.666667     0.9527025           1.333333   10.03770  Within ADL
400     -2.699531      -2.166667    -0.6739850          -0.541667   20.02664  Within ADL
800     -1.223772      -1.133333    -0.1529542          -0.141667   40.00452  Within ADL
1600    -2.305587      -3.100000    -0.1441708          -0.193750   79.96028  Within ADL
3200    10.364116       7.800000     0.3241384           0.243750  159.87179  Within ADL
6400    -4.329810     -10.433333    -0.0677179          -0.163021  319.69482  Within ADL

Definitions

Residual = observed mean - predicted value.
Recovery bias = observed mean - expected value.
Recovery bias (%) = 100 * recovery bias / expected value.
Residual (%) denominator: predicted.
ADL decision metric = residual.
Code 
plot(fit_lin, type = "fit")

Code 
plot(fit_lin, type = "residual")

adl_exceeded is a numeric comparison with the ADL supplied by the caller, not a complete product-acceptance decision. Choose weights according to the design and heteroscedasticity structure, and compare residuals, back-calculated bias, and prespecified criteria.

14.3.2 Polynomial Sensitivity Analysis

Code 
poly_data <- read_csv("./data/014-线性分析/linearity-polynomial.csv", show_col_types = FALSE)
poly_data <- as.data.frame(poly_data)
head(poly_data, 6)
    conc replicate signal
1  75.00         1   75.0
2  75.00         2   75.0
3  75.00         3   74.4
4 483.75         1  454.1
5 483.75         2  455.3
6 483.75         3  456.0
Code 
fit_poly <- linearity(
  poly_data, sample = "conc", result = "signal", x = "conc",
  method = "polynomial", weights = "level_variance", max_degree = 3,
  poly_alpha = 0.05, percent_base = "predicted"
)
print(fit_poly)
Linearity regression analysis

Analysis settings

Method: polynomial (EP6-A, exploratory)
Observations / levels: 15 / 5
Intercept: estimated
Weights: level_variance
Fit unit: observation
Selected degree: 3

Model comparison

Degree: 1
Observations                : 15
Parameters                  : 2
Residual degrees of freedom : 13
RSS                         : 52631.917
Weighted RSS                : 550.93015
SY.X                        : 63.62868
Weighted SY.X               : 6.5099341
R-squared                   : 0.99902344
Adjusted R-squared          : 0.99894832

Degree: 2
Observations                : 15
Parameters                  : 3
Residual degrees of freedom : 12
RSS                         : 10241.163
Weighted RSS                : 30.175825
SY.X                        : 29.213529
Weighted SY.X               : 1.5857655
R-squared                   : 0.99994651
Adjusted R-squared          : 0.9999376

Degree: 3
Observations                : 15
Parameters                  : 4
Residual degrees of freedom : 11
RSS                         : 4360.5146
Weighted RSS                : 15.881083
SY.X                        : 19.910058
Weighted SY.X               : 1.2015552
R-squared                   : 0.99997185
Adjusted R-squared          : 0.99996417
Selected                    : Yes

Regression coefficients (degree 1)

    Estimate Std. Error   t value Pr(>|t|)    
a 3.26682147 1.60149874   2.03985 0.062224 .  
b 0.94635339 0.00820623 115.32135  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Regression coefficients (degree 2)

     Estimate  Std. Error  t value   Pr(>|t|)    
a 7.07723e+00 4.71485e-01  15.0105 3.8555e-09 ***
b 8.98365e-01 3.88796e-03 231.0636 < 2.22e-16 ***
c 5.87581e-05 4.08310e-06  14.3905 6.2393e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Regression coefficients (degree 3, selected)

      Estimate   Std. Error  t value   Pr(>|t|)    
a  1.08167e+01  1.24094e+00  8.71655 2.8637e-06 ***
b  8.38991e-01  1.90976e-02 43.93179 1.0374e-13 ***
c  1.93013e-04  4.27784e-05  4.51193  0.0008837 ***
d -5.97186e-08  1.89786e-08 -3.14662  0.0092980 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Sample fit information

Sample   Expected  Predicted  Observed mean      Weight  Observed SD  n
75          75.00    74.8015         74.800  8.33333333     0.346410  3
483.75     483.75   455.0861        455.133  1.08303249     0.960902  3
892.5      892.50   870.9067        865.967  0.01552474     8.025792  3
1301.25   1301.25  1297.7933       1328.467  0.00166688    24.493332  3
1710      1710.00  1711.2760       1710.433  0.01516914     8.119319  3

Sample bias information

Sample      Residual  Recovery bias  Residual (%)  Recovery bias (%)         DL  ADL  Decision    
75       -0.00153386      -0.200000   -0.00206599         -0.2666667   0.558208   NA  Not assessed
483.75    0.04720887     -28.616667    0.01023909         -5.9155900  -5.979149   NA  Not assessed
892.5    -4.94005897     -26.533333   -0.58263161         -2.9729225  23.019504   NA  Not assessed
1301.25  30.67332481      27.216667    2.48425504          2.0915786  63.084173   NA  Not assessed
1710     -0.84264426       0.433333   -0.05196596          0.0253411  89.744861   NA  Not assessed

Definitions

Residual = observed mean - predicted value.
Recovery bias = observed mean - expected value.
Recovery bias (%) = 100 * recovery bias / expected value.
Residual (%) denominator: predicted.
ADL decision metric = deviation from linearity (DL).

Warnings / interpretation notes

1 duplicate sample/result/x row(s) were retained.

Polynomial regression follows superseded EP6-A and is exploratory under EP06-Ed2.
Code 
plot(fit_poly, type = "fit")

Code 
plot(fit_poly, type = "difference")

14.3.3 Standard Cross-Checks

The package can reproduce the principal calculations in CLSI EP06 Appendices G–I and the YY/T 1789.4 Appendix A.5 linearity example using the supplied CSV files. The following pattern illustrates the EP06 Appendix G weighting and the reporting convention:

Code 
ep06_g <- as.data.frame(read_csv(
  "./data/014-线性分析/std-ep06-appendix-g.csv", show_col_types = FALSE
))
ep06_g_ivd <- linearity(
  ep06_g, sample = "level", result = "result", x = "expected",
  method = "linear", intercept = FALSE, weights = "pooled",
  variance_group = "variance_group"
)
ep06_g_ivd
Linearity regression analysis

Analysis settings

Method: linear (EP06-Ed2)
Observations / levels: 20 / 10
Intercept: fixed at zero
Weights: pooled
Variance group: variance_group
Fit unit: level_mean

Model comparison

Degree: 1
Observations                : 10
Parameters                  : 1
Residual degrees of freedom : 9
RSS                         : 2010.7972
Weighted RSS                : 83.183953
SY.X                        : 14.947304
Weighted SY.X               : 3.0401746
R-squared                   : 0.99911786
Adjusted R-squared          : 0.99901984

Regression coefficients

   Estimate Std. Error t value   Pr(>|t|)    
b 1.0677279  0.0105755 100.962 4.6544e-15 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Pooled variance groups

Variance group: low
Levels             : 3
Degrees of freedom : 3
Minimum x          : 52.9
Maximum x          : 158.7
Pooled variance    : 1.21
Pooled SD          : 1.1
Level-mean weight  : 1.6528926
Minimum weight     : 1.6528926
Maximum weight     : 1.6528926

Variance group: middle
Levels             : 4
Degrees of freedom : 4
Minimum x          : 211.6
Maximum x          : 370.3
Pooled variance    : 55.255
Pooled SD          : 7.4333707
Level-mean weight  : 0.036195819
Minimum weight     : 0.036195819
Maximum weight     : 0.036195819

Variance group: high
Levels             : 3
Degrees of freedom : 3
Minimum x          : 423.2
Maximum x          : 529
Pooled variance    : 267.72167
Pooled SD          : 16.362202
Level-mean weight  : 0.0074704451
Minimum weight     : 0.0074704451
Maximum weight     : 0.0074704451

Sample fit information

Sample  Expected  Predicted  Observed mean      Weight  Observed SD  n
10          52.9    56.4828          51.25  1.65289256     0.353553  2
9          105.8   112.9656         111.30  1.65289256     1.838478  2
8          158.7   169.4484         171.95  1.65289256     0.353553  2
7          211.6   225.9312         233.15  0.03619582     5.586144  2
6          264.5   282.4140         292.05  0.03619582     6.293250  2
5          317.4   338.8968         350.15  0.03619582     4.879037  2
4          370.3   395.3796         391.45  0.03619582    11.242998  2
3          423.2   451.8624         468.20  0.00747045    20.364675  2
2          476.1   508.3453         496.65  0.00747045    18.879751  2
1          529.0   564.8281         529.00  0.00747045     5.656854  2

Sample bias information

Sample   Residual  Recovery bias  Residual (%)  Recovery bias (%)  ADL  Decision    
10       -5.23281          -1.65      -9.26442           -3.11909   NA  Not assessed
9        -1.66561           5.50      -1.47444            5.19849   NA  Not assessed
8         2.50158          13.25       1.47631            8.34909   NA  Not assessed
7         7.21878          21.55       3.19512           10.18431   NA  Not assessed
6         9.63597          27.55       3.41200           10.41588   NA  Not assessed
5        11.25317          32.75       3.32053           10.31821   NA  Not assessed
4        -3.92964          21.15      -0.99389            5.71159   NA  Not assessed
3        16.33755          45.00       3.61560           10.63327   NA  Not assessed
2       -11.69525          20.55      -2.30065            4.31632   NA  Not assessed
1       -35.82806           0.00      -6.34318            0.00000   NA  Not assessed

Definitions

Residual = observed mean - predicted value.
Recovery bias = observed mean - expected value.
Recovery bias (%) = 100 * recovery bias / expected value.
Residual (%) denominator: predicted.
ADL decision metric = residual.

The ten levels are explicitly grouped as 1–3, 4–7, and 8–10. The user supplies these groups from the repeatability-SD–concentration assessment and protocol; the function does not select a favorable grouping.

For Appendix H, use weights = "level_variance" and fit the combined panel and each panel separately. For Appendix I, use the published summary data and weights \(10/s^2\) rather than manufacturing pseudo-raw replicates. The standard’s higher-order model-selection and ADL-PctBnd lookup are not identical to the package’s exploratory polynomial mode; report this algorithm difference explicitly.

Code 
yyt1789_4 <- read.csv("./data/014-线性分析/std-ep06-appendix-g.csv")
yyt1789_4_fit <- linearity(
  yyt1789_4, sample = "level", result = "result", x = "expected",
  method = "linear", weights = "equal", adl_rel = 10
)
yyt1789_4_fit
Linearity regression analysis

Analysis settings

Method: linear (EP06-Ed2)
Observations / levels: 20 / 10
Intercept: estimated
Weights: equal
Fit unit: level_mean

Model comparison

Degree: 1
Observations                : 10
Parameters                  : 2
Residual degrees of freedom : 8
RSS                         : 1709.9271
Weighted RSS                : 1709.9271
SY.X                        : 14.61988
Weighted SY.X               : 14.61988
R-squared                   : 0.99309927
Adjusted R-squared          : 0.99223668

Regression coefficients

   Estimate Std. Error t value   Pr(>|t|)    
a 9.1333333  9.9872792  0.9145    0.38719    
b 1.0324168  0.0304271 33.9308 6.2178e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Sample fit information

Sample  Expected  Predicted  Observed mean  Weight  Observed SD  n
10          52.9    63.7482          51.25       1     0.353553  2
9          105.8   118.3630         111.30       1     1.838478  2
8          158.7   172.9779         171.95       1     0.353553  2
7          211.6   227.5927         233.15       1     5.586144  2
6          264.5   282.2076         292.05       1     6.293250  2
5          317.4   336.8224         350.15       1     4.879037  2
4          370.3   391.4373         391.45       1    11.242998  2
3          423.2   446.0521         468.20       1    20.364675  2
2          476.1   500.6670         496.65       1    18.879751  2
1          529.0   555.2818         529.00       1     5.656854  2

Sample bias information

Sample     Residual  Recovery bias  Residual (%)  Recovery bias (%)       ADL  Decision   
10      -12.4981818          -1.65  -19.60555025           -3.11909   6.37482  Exceeds ADL
9        -7.0630303           5.50   -5.96726046            5.19849  11.83630  Within ADL 
8        -1.0278788          13.25   -0.59422557            8.34909  17.29779  Within ADL 
7         5.5572727          21.55    2.44176200           10.18431  22.75927  Within ADL 
6         9.8424242          27.55    3.48765416           10.41588  28.22076  Within ADL 
5        13.3275758          32.75    3.95685524           10.31821  33.68224  Within ADL 
4         0.0127273          21.15    0.00325142            5.71159  39.14373  Within ADL 
3        22.1478788          45.00    4.96531184           10.63327  44.60521  Within ADL 
2        -4.0169697          20.55   -0.80232369            4.31632  50.06670  Within ADL 
1       -26.2818182           0.00   -4.73305938            0.00000  55.52818  Within ADL 

Definitions

Residual = observed mean - predicted value.
Recovery bias = observed mean - expected value.
Recovery bias (%) = 100 * recovery bias / expected value.
Residual (%) denominator: predicted.
ADL decision metric = residual.

14.4 Interpretation

  1. A successful fit does not establish model suitability; evaluate residuals, bias, acceptance limits, and the protocol.
  2. Examine residuals on both original and relative scales, especially when values span orders of magnitude.
  3. Select weights using the observed variance structure or protocol requirements.
  4. Conclusions apply only within the validated range; avoid unsupported extrapolation.
  5. Preserve raw data, code, model parameters, convergence information, package version, and session information.

14.5 Complete Standard Cases

14.5.1 CLSI EP06 Second Edition Appendix G: Two-Replicate Study

Appendix G prepares ten levels by mixing HIGH with blank at relative concentrations of 0.1–1.0, with two measurements per level. The prespecified linear interval is 55–500 U/L and the allowable deviation from linearity is ±10%. Because there are only two replicates per level, the standard pools adjacent-level variances into three groups and uses \(2/\overline{s^2}\) as the no-intercept WLS weight. The grouping column is written directly into the CSV according to standard Table G3, and expected values are already calculated from the HIGH concentration and preparation proportions.

Code 
ep06_g <- as.data.frame(read_csv(
  "./data/014-线性分析/std-ep06-appendix-g.csv",
  show_col_types = FALSE
))
Code 
ep06_g_ivd <- linearity(
  ep06_g, sample = "level", result = "result", x = "expected",
  method = "linear", intercept = FALSE,
  weights = "pooled", variance_group = "variance_group"
)
EP06 Second Edition Appendix G: ivdtools compared with the standard text
Standard location Statistic Standard ivdtools
EP06 Ed2 Appendix G WLS_slope 1.0677 1.0677
EP06 Ed2 Appendix G max_abs_deviation_percent 9.3000 9.2644

variance_group explicitly assigns levels 1–3, 4–7, and 8–10 to three groups. The grouping is supplied by the analyst from the repeatability-SD versus concentration plot and the prespecified plan; the function does not select the most favorable grouping automatically.

14.5.2 CLSI EP06 Second Edition Appendix H: Two-Panel Study

Appendix H prepares two panels, HIGH1 and HIGH2, separately with blank, at ten levels per panel and five measurements per level, for 100 results. HIGH1 covers the high end and HIGH2 the low end; together they cover the case’s prespecified 20–1,000 U/L range. Each level receives weight \(5/s^2\) in a no-intercept WLS fit. The CSV includes sample_id and expected values determined by panel, level, and preparation proportion, so it can enter the analysis directly.

Code 
ep06_h <- as.data.frame(read_csv(
  "./data/014-线性分析/std-ep06-appendix-h.csv",
  show_col_types = FALSE
))
Code 
ep06_h_fit <- function(z) {
  fit <- linearity(
    z, sample = "sample_id", result = "result", x = "expected",
    method = "linear", intercept = FALSE, weights = "level_variance"
  )
  data.frame(
    slope = fit$coefficients$estimate,
    max_abs_deviation_percent = max(abs(fit$level_summary$percent_residual_linear))
  )
}
ep06_h_combined <- ep06_h_fit(ep06_h)
ep06_h_separate <- do.call(rbind, lapply(
  split(ep06_h, ep06_h$panel), ep06_h_fit
))
EP06 Second Edition Appendix H: ivdtools compared with the standard text
Standard location Statistic Standard ivdtools
EP06 Ed2 Appendix H combined WLS slope 0.9686 0.9686
EP06 Ed2 Appendix H combined maximum deviation (%) 14.5000 14.4759
EP06 Ed2 Appendix H HIGH1 WLS slope 0.9496 0.9496
EP06 Ed2 Appendix H HIGH1 maximum deviation (%) 12.8000 12.7612
EP06 Ed2 Appendix H HIGH2 WLS slope 1.0013 1.0013
EP06 Ed2 Appendix H HIGH2 maximum deviation (%) 9.2000 9.2134

The maximum deviation in all three analyses is below the case’s prespecified ±15%, but the overlap, range coverage, and weighting structure of the two panels are prerequisites for that conclusion.

14.5.3 CLSI EP06 Second Edition Appendix I: Assigned Samples

Appendix I provides true values, ten-measurement means, and SDs for nine spiked samples but does not publish the 90 individual measurements. This case therefore retains the published summary granularity and does not create pseudo-raw observations from the means and SDs. The weight is \(10/s^2\), the fitted model is a no-intercept WLS line, and the prespecified allowable deviation is ±2%.

Code 
ep06_i <- as.data.frame(read_csv(
  "./data/014-线性分析/std-ep06-appendix-i-summary.csv",
  show_col_types = FALSE
))
ep06_i$weight <- ep06_i$n / ep06_i$sd^2
ep06_i_ivd <- linearity(
  ep06_i, sample = "sample", result = "measured_mean", x = "true_value",
  method = "linear", intercept = FALSE, weights = ep06_i$weight
)
EP06 Second Edition Appendix I: ivdtools compared with the standard text
Standard location Statistic Standard ivdtools
EP06 Ed2 Appendix I WLS_slope 1.0045 1.0045
EP06 Ed2 Appendix I max_abs_deviation_percent 1.7000 1.7212

The recalculated maximum absolute deviation is within the case’s prespecified ±2%. Recovery bias and linearity deviation answer different questions: the former is relative to the true value, while the latter is relative to the best-fit line, so they cannot replace one another.

14.5.4 WS/T 408–2024: Five-Level Linearity Verification

The HDL-C case in Appendix A.4 contains five levels with three measurements per level. The standard focuses on the residual SD of the straight-line regression, \(s_{y|x}\), the within-run SD \(s_{WR}\), their F ratio, and the nonlinear SD \(s_{NL}\).

Code 
wst408_linearity <- read_csv(
  "./data/014-线性分析/std-wst408-linearity.csv",
  show_col_types = FALSE
)

wst408_linearity_ivd <- linearity(
  as.data.frame(wst408_linearity), sample = "level", result = "measured",
  x = "expected", method = "linear", conf.level = 0.95
)
WS/T 408 Appendix A.4: ivdtools compared with the standard text
Standard location Statistic Standard ivdtools
WS/T 408 Appendix A.4 s_yx 0.0318 0.03277
WS/T 408 Appendix A.4 s_WR 0.0186 NA
WS/T 408 Appendix A.4 F 2.9130 NA
WS/T 408 Appendix A.4 s_NL 0.0258 NA

linearity() fits the line to the five level means and returns \(s_{y|x}\approx0.0328\); the standard companion table gives 0.0318. The s_WR, F, and s_NL quantities required by the standard are not public results in the current object, so the table marks them explicitly as NA.

14.5.5 YY/T 1789.4–2022 Appendix A: Linearity Sample 1

Appendix A.5 provides 14 dilution levels for AFP linearity sample 1, with four results per level. The standard selects a model using significance of higher-order polynomial terms and then evaluates level-specific relative bias from the final line against a ±10% allowable error.

Code 
yyt1789_4 <- read.csv(
  "./data/014-线性分析/std-yyt1789-4-appendix-a-sample1.csv"
)
yyt1789_4_linear_p <- linearity(
  yyt1789_4, sample = "level", result = "result", x = "dilution",
  method = "polynomial", weights = "equal", adl_rel = 10
)
yyt1789_4_linear <- linearity(
  yyt1789_4, sample = "level", result = "result", x = "dilution",
  method = "linear", weights = "equal", adl_rel = 10
)
YY/T 1789.4–2022 Appendix A.5: final line and relative bias
Standard location Statistic Standard ivdtools
YY/T 1789.4 Appendix A.5 / level 1 final-line prediction 4.28 4.284
YY/T 1789.4 Appendix A.5 / level 1 relative bias (%) 6.07 5.974
YY/T 1789.4 Appendix A.5 / level 2 final-line prediction 19.40 19.401
YY/T 1789.4 Appendix A.5 / level 2 relative bias (%) 5.05 5.046
YY/T 1789.4 Appendix A.5 / level 3 final-line prediction 34.52 34.518
YY/T 1789.4 Appendix A.5 / level 3 relative bias (%) 4.46 4.453
YY/T 1789.4 Appendix A.5 / level 4 final-line prediction 64.75 64.752
YY/T 1789.4 Appendix A.5 / level 4 relative bias (%) -3.69 -3.697
YY/T 1789.4 Appendix A.5 / level 5 final-line prediction 125.22 125.219
YY/T 1789.4 Appendix A.5 / level 5 relative bias (%) 7.25 7.250
YY/T 1789.4 Appendix A.5 / level 6 final-line prediction 246.16 246.154
YY/T 1789.4 Appendix A.5 / level 6 relative bias (%) 0.47 0.467
YY/T 1789.4 Appendix A.5 / level 7 final-line prediction 367.10 367.089
YY/T 1789.4 Appendix A.5 / level 7 relative bias (%) 3.77 3.773
YY/T 1789.4 Appendix A.5 / level 8 final-line prediction 488.04 488.025
YY/T 1789.4 Appendix A.5 / level 8 relative bias (%) -4.04 -4.043
YY/T 1789.4 Appendix A.5 / level 9 final-line prediction 608.98 608.960
YY/T 1789.4 Appendix A.5 / level 9 relative bias (%) -3.05 -3.051
YY/T 1789.4 Appendix A.5 / level 10 final-line prediction 729.92 729.895
YY/T 1789.4 Appendix A.5 / level 10 relative bias (%) 3.59 3.598
YY/T 1789.4 Appendix A.5 / level 11 final-line prediction 850.86 850.830
YY/T 1789.4 Appendix A.5 / level 11 relative bias (%) -3.99 -3.990
YY/T 1789.4 Appendix A.5 / level 12 final-line prediction 971.80 971.765
YY/T 1789.4 Appendix A.5 / level 12 relative bias (%) 2.60 2.607
YY/T 1789.4 Appendix A.5 / level 13 final-line prediction 1092.74 1092.700
YY/T 1789.4 Appendix A.5 / level 13 relative bias (%) -3.20 -3.198
YY/T 1789.4 Appendix A.5 / level 14 final-line prediction 1213.68 1213.635
YY/T 1789.4 Appendix A.5 / level 14 relative bias (%) 2.56 2.566

The model-selection procedures are not identical: the standard fits polynomials to level means and reports regression standard errors, whereas linearity(method = “polynomial”) currently fits individual observations and labels this mode as the legacy EP6-A exploratory workflow; the degrees of freedom and regression standard errors differ. This case therefore calls the package’s linear method a reproduction of the final line only and does not present its polynomial selection as YY/T 1789.4’s higher-order significance test.

The ADL-PctBnd critical table in standard Appendix A.3 also combines regression imprecision, number of levels, and replicate count. adl_rel = 10 is a level-wise numerical limit and does not implement the PctBnd lookup rule. The two similarly named ADL concepts use different algorithms and must be distinguished in reporting.