Gabay

Explanations

Models - explanations

Why linear model?

  • ISO 5725 outlines various sources of variability in measurement methods, including the operator, equipment, calibration, environment, and time elapsed between measurements.

  • In our study, we consider fixed factors such as the type of device, tooth number, orientation, direction, and part of the tooth, to investigate their effects on trueness and precision estimation in dental measurements.

  • ISO 5725 provides detailed information about the linear model and its application for trueness and precision estimation in measurement methods. For specific details and guidelines on implementing the linear model, refer to Section 4.4 of ISO-5725-5.

Here are the updated model specifications:

  1. Full Model:
    The most detailed model includes all of the fixed effects. The formula for this model is:
    \[y = \beta_0 + \beta_1x_{\text{device}} + \beta_2x_{\text{tooth}} + \beta_3x_{\text{orientation}} + \beta_4x_{\text{direction}} + \beta_5x_{\text{part}} + b_{\text{model}} + \epsilon\]

  2. Aggregated Location Model:
    This model assumes that the location does not significantly impact the measurement values, and thus omits this variable:
    \[y = \beta_0 + \beta_1x_{\text{device}} + \beta_2x_{\text{tooth}} + \beta_3x_{\text{orientation}} + \beta_4x_{\text{direction}} + b_{\text{model}} + \epsilon\]

  3. Aggregated Orientation Model:
    Here, we explore the possibility that the orientation does not significantly influence the measurement values. The formula for this model is:
    \[y = \beta_0 + \beta_1x_{\text{device}} + \beta_2x_{\text{tooth}} + \beta_3x_{\text{direction}} + \beta_4x_{\text{part}} + b_{\text{model}} + \epsilon\]

  4. Aggregated Location and Orientation Model:
    The simplest model aggregates over both the ‘location’ and ‘orientation’ variables. Its formula is:
    \[y = \beta_0 + \beta_1x_{\text{device}} + \beta_2x_{\text{tooth}} + \beta_3x_{\text{direction}} + b_{\text{model}} + \epsilon\]

In these formulas, \(y\) represents the outcome variable, \(x_{\text{...}}\) denote the fixed effect covariates, \(\beta_{\text{...}}\) are the coefficients of these fixed effects, \(b_{\text{model}}\) is the random intercept for each model, and \(\epsilon\) is the error term.

The general form of the linear model is:

\[ Y = X\beta + U\zeta + \varepsilon\] Where \(X\) are the fixed-effect data (all variables you have in the data, except the model); \(U\) are the random effects (here it is just the indicator of model 1-16). We estimate \(\beta\) with the device type as a covariate.

What does this have to do with your paper???

  • Inference on trueness - done through the coefficient of the device indicator.

  • Inference on precision - comparing the error term from the linear model that accounts for the error.

Estimation

Trueness

All four distances aggregated

model ci
Full [-0.031 - 0.0365]
Aggregated location [-0.0313 - 0.0368]
Aggregated orientation [-0.0315 - 0.037]
Aggregated both [-0.0318 - 0.0373]

$Full

term estimate std.error statistic
(Intercept) -0.4950215 0.0842209 -5.8776581
model 0.0000370 0.0053979 0.0068535
tooth 0.0422070 0.0034490 12.2372922
devicesb 0.0027539 0.0172452 0.1596907
locationneck -0.0737695 0.0172452 -4.2776726
orientationvertical 0.0968555 0.0172452 5.6163565
directionmesial -0.0071289 0.0172452 -0.4133838
sd__(Intercept) 0.0933660 NA NA
sd__Observation 0.2759240 NA NA

$Aggregated location

term estimate std.error statistic
(Intercept) -0.5319062 0.0842187 -6.3157705
model 0.0000370 0.0053979 0.0068535
tooth 0.0422070 0.0034786 12.1332136
devicesb 0.0027539 0.0173932 0.1583325
orientationvertical 0.0968555 0.0173932 5.5685892
directionmesial -0.0071289 0.0173932 -0.4098679
sd__(Intercept) 0.0932561 NA NA
sd__Observation 0.2782909 NA NA

$Aggregated orientation

term estimate std.error statistic
(Intercept) -0.4465937 0.0845544 -5.2817316
model 0.0000370 0.0053979 0.0068535
tooth 0.0422070 0.0035011 12.0552943
devicesb 0.0027539 0.0175056 0.1573157
locationneck -0.0737695 0.0175056 -4.2140534
directionmesial -0.0071289 0.0175056 -0.4072358
sd__(Intercept) 0.0931720 NA NA
sd__Observation 0.2800896 NA NA

$Aggregated both

term estimate std.error statistic
(Intercept) -0.4834785 0.0845302 -5.7195975
model 0.0000370 0.0053979 0.0068535
tooth 0.0422070 0.0035302 11.9560239
devicesb 0.0027539 0.0176509 0.1560203
directionmesial -0.0071289 0.0176509 -0.4038823
sd__(Intercept) 0.0930622 NA NA
sd__Observation 0.2824152 NA NA

Precision

Confindece intervals for precision under various models:

Aggregation SB MFA
Aggregate location [0.2402 - 0.2977] [0.2635 - 0.3111]
Aggregate location and orientation [0.2425 - 0.3022] [0.2668 - 0.3169]
Aggregate orientation [0.2403 - 0.2984] [0.2648 - 0.3135]
No aggregation [0.2381 - 0.2936] [0.2622 - 0.3082]

Literature

I reviewed the papers you sent. Here are the highlights relevant to the current analysis.

In page 34 @fluegge2017 write “The relation between each measurement and the system (S1 and S2) was evaluated with a nonparametric test as in Brunner et al.”. They are hiding behind a heavy methodological reference, without explicitly stating the methods of analysis!!! Even I don’t do such things…

The paper by [@mangano2016] is actually of worth, as they report their statistical analysis with detail.

@imburgia2017 rely on the same method as @mangano2016.

We use the VCA R package [@VCA].

$tooth.21


Result Variance Component Analysis:
-----------------------------------

  Name                     DF        SS        MS       VC       %Total   
1 total                    38.173963                    1.683879 100      
2 device                   1         2.911289  2.911289 0.079387 4.714536 
3 model                    15        26.309898 1.753993 0*       0*       
4 model:replication_number 16        45.409325 2.838083 1.23359  73.258831
5 error                    31        11.497961 0.370902 0.370902 22.026634
  SD       CV[%]    
1 1.297644 85.169936
2 0.281757 18.492932
3 0*       0*       
4 1.110671 72.898117
5 0.609017 39.972415

Mean: 1.523594 (N = 64) 

Experimental Design: balanced  |  Method: ANOVA | * VC set to 0 | adapted MS used for total DF


$tooth.16


Result Variance Component Analysis:
-----------------------------------

  Name                     DF        SS        MS       VC       %Total   
1 total                    35.288206                    0.65227  100      
2 device                   1         0.49      0.49     0.008439 1.293807 
3 model                    15        23.480394 1.56536  0.248823 38.147338
4 model:replication_number 16        9.12105   0.570066 0.175059 26.838383
5 error                    31        6.8184    0.219948 0.219948 33.720471
  SD       CV[%]    
1 0.807632 86.988311
2 0.091865 9.894543 
3 0.498822 53.727052
4 0.4184   45.064969
5 0.468987 50.51353 

Mean: 0.928438 (N = 64) 

Experimental Design: balanced  |  Method: ANOVA

Attempt to use a linear model via the standard R package for linear modeling lme4 [@lme4].

Questions

  1. In response to the following reviewer remark:

    The evaluation is wrong. The planned position is defined as a reference. However, this is deviation from that due to implant placement error. The reference position should be real implant position, obtained by industrial or model scanner, and deviations on scans, obtained with regular scan body and FMA should be measured and compared.

    You wrote (in answers 29.04.23):

    Answer: we added another set of measurements, in which, laboratory 3D scanner was used for SBIO models scanning. These model scans were used as a reference for implants positions when compared to the MFA and SBIO intraoral scans.

    Does this imply that the absolute errors you provided me with are (for a given device, tooth and model) of the form: \[|\text{Actual implant location} - \text{Measured implant location} |\]?

The reviewers seem insistent on this issue, as they all follow @mangano2016.

  1. Somewhat related to question 1 above, can we obtain data in the form of the actual value (not only the deviation?).
  2. Does it make sense to average out mesial and distal measurements? Or use them as additional replications?
  3. I do not understand why @mangano2016 reports obtaining estimates from ANOVA…