Statistical Methodology for Medical Data: Gestational Age,Birth Weight,and Hand Characteristics

Abstract

We briefly review the current trend in handling the neonatal baby weight, gestational age, and other related categorical and continuous variable data, through both robust and outlier-based methods. Graphs are generally presented as smooth curves which give a wrong visual impression and totally hide the actual variability of the data, robust methods are applied without considering that they may certainly be affected by asymmetrical contamination, and the outlier-based methods are incorrectly applied without ascertaining if the individual data arrays are normally distributed, free from discordant observations.

Aim: To demonstrate our new procedure for handling of neonatal baby data from two case studies.

Methodology: The univariate data were handled through the application of discordancy tests in the light of new precise and accurate critical values, and significance tests were applied to both original and discordant outlier-free data arrays. The bivariate data were also handled from identifying and separating discordant outliers and reporting the quality of the coefficients in terms of standard errors. Finally, the supervised multivariate technique of linear discriminant and canonical analysis is successfully applied.

Results: The results of significance tests for some cases varied from the use of all data versus censored data. The identification and separation of bivariate discordant outliers drastically improved the quality of regression lines. Linear discriminant and canonical analysis provided high success values for laboratory-controlled experiments.

Conclusion: The need of obtaining both central tendency and dispersion parameters in all medical data is established. The reporting of regression coefficients with individual errors is recommended.

Keywords

Univariate data analysis system bivariate data computer programs ordinary linear regression discordancy tests

Introduction

The relationship and statistical treatment of neonatal baby birth weight and other characteristics with gestational age and other explanatory factors has been the subject of research at least since 1900^1-4 and is so even today.^5-8 The emphasis of such studies has been on charts of birth weights and gestational age as well as graphs of actual and smooth percentile curves.^8-20 The central tendency and dispersion parameters were often represented by the mean and standard deviation, respectively, without any prior statistical treatment of the data.^{3, 4, 6, 11-14, 19-27} Linear and polynomial regression equations were also reported in some studies.^{3, 4, 6, 14, 26, 27} Diagrams showing bivariate relationships were presented without reporting any regression equations.^{5, 15, 16, 18-24, 28} Statistical tests, such as the t test, were applied in some studies to draw inferences.^{14, 17, 23, 24}

The main deficiency concerning the statistical treatment of these data seems to be the lack of application of modern discordancy tests^29-37 in the handling of univariate, bivariate, or multivariate data, because no commercial software is available for this purpose. However, appropriate software, such as UDASys2 (Univariate Data Analysis System version 2),³⁴ UDASys3 (Univariate Data Analysis System version 3),³⁵ BIDASys (Bivariate Data Analysis System),³⁶ and DOMuDaF (Discordant Outliers from Multivariate Data from the F test of w),³⁷ have been recently developed and published to facilitate these applications.

Not only the mean and standard deviation values are affected by the presence of discordant outliers but also the so-called robust parameters such as the median, including the percentiles, are prone to them.^{34, 35, 38-40} In one study,¹¹ outlying observations were identified but not from the modern tests based on new precise and accurate critical values and those having the best performance (highest power of test and lowest masking and swamping effects).^{29, 33-40}

The other problem is with the reporting of regression equations, in which the coefficients are dealt with as constants, i.e., without any error or uncertainty.^{3, 4, 6, 14, 26, 27} Commercial software, such as Statistica, and freely available programs, such as BiDASys,³⁶ can provide the standard errors or uncertainties on these coefficients.

In this work, we briefly comment on the computer programs available at the web portal http://tlaloc.ier.unam.mx and illustrate our refined statistical procedure from 2 case studies. None of the papers^1-28 could be used for illustration, because the original data were not available in any of them.

Recent Advances in Statistical Handling of Data

We emphasize that the univariate data are not being handled correctly, because the outlier-based central tendency (mean) and dispersion (standard deviation) parameters are used, without considering the fact that the mean and standard deviation belong to the category of outlier-based methods and are highly influenced by the presence of outlying observations.^{29, 39, 40} The application of the significance tests (F, t, and analysis of variance [ANOVA]) also requires that the individual statistical samples under evaluation be normally distributed, free from discordant observations. It is of utmost importance that discordant observations be objectively identified and separated by appropriate statistical procedures.²⁹ The reasons for discordancy should be sought for.

Both versions of a computer program UDASys (UDASys2³⁴ and UDASys3³⁵) apply recursive tests^41-43 with prior application of single-outlier discordancy tests,^{29, 44, 45} all with new, precise, and accurate critical values.^{34, 35, 46-49} The application of significance tests (F, t, and ANOVA) also requires the normality of the statistical samples, which can be achieved through discordancy tests from any of the 2 versions of UDASys, prior to the inference about similarities or differences between or among statistical samples. Thus, the discordancy tests can be applied to samples of sizes as large as 30,000 and significance tests to the number of sample groups as large as 30. The main difference between the 2 versions of UDASys is that UDASys3 is more automatized than UDASys2 for the handling of multivariate data of geochemical reference materials.

Bivariate data can be adequately handled from the online program BiDASys.³⁶ This program provides an innovation through the identification and separation of bivariate outliers²⁹ for bivariate data arrays as large as 1,000 and the provision of uncertainty-weighted linear regression (UWLR), although the latter can be applied only when errors on the individual data are available. Unfortunately, this is not the case in most studies in medical research because individual errors or uncertainties are seldom reported.

The handling of multivariate data could also be benefitted from the validity of the assumption of multinormality,⁵⁰ which can be achieved from DOMuDaF³⁷ in sample sizes up to 30,000.

We will highlight the use of these programs in 2 case studies. To reduce type I error, all tests will be applied at the high confidence level of 99%,^{34, 35} rather than the commonly used 95%.⁵¹

The First Case Study

The North Carolina medical data were downloaded from the website https://www.openintro.org/stat/data/. The downloaded Excel file consisted of 1,000 random samples (rows) and 14 variables (columns; 9 categorical and 5 continuous). Because there were empty cells, all data could not be used. Only the birth of a singleton baby was considered. Individual groups were established for handling of the data. The programs used for data handling also provided some graphs, but we decided not to present them to keep the paper short.

Univariate Data

We divided the database into 2 sets, each of 2 groups (Gr1-Gr2 and Gr3-Gr4; Table 1) and then used UDASys to interpret the data. Univariate outlying observations were identified and separated from each group and the remaining censored outlier-free data (n_male = 468; n_female = 473; Table 1) were processed in UDASys. The mean $\bar{x}$ values and the dispersion parameters (standard deviation s, standard error of the mean se, and uncertainty of the mean at the 99% confidence limits u₉₉) as well as the 99% confidence interval of the mean CI₉₉ values are included in Table 1. The CI₉₉ values of the weight of mail babies (3355-3479 g) and female babies (3170-3280 g). The t test at the 99% confidence level revealed that the male babies have higher weight than the female babies at the 99% confidence level (Table 1). Similarly, the Gr3 to Gr4 subdivision showed that the babies born to nonsmoking mothers have higher weights (3288-3380 g versus 3038-3285 g) than those born to smoking mothers (Table 1).

We then subdivided the data into 6 groups (Gr5 to Gr10) and processed them in UDASys (Table 2). We decided to process both sets of data (original and discordant outlier-free data) to highlight the importance of discordancy or fulfillment of the normality assumption. The following inferences were drawn from Student t test (Table 2): Gr5 to Gr6 male babies born to nonsmoking white mothers have higher weight than those born to nonsmoking black mothers, for both sets; Gr7 to Gr8 female babies born to nonsmoking white mothers have higher weight than those born to nonsmoking black mothers, for both sets; Gr9 to Gr10 male babies born to smoking white mothers have more weight than female babies for discordant outlier-free data but not for the original data. Thus, the fulfillment of the assumption of normality changed the inference of the t test for the group pair Gr9 to Gr10.

Other combinations of tests are also possible, eg, Gr5 can be compared with other groups Gr7 to Gr10 and statistical inference can be drawn. For example, Gr5 to Gr7 would show that male babies born to nonsmoking white mothers have higher weight than female babies born to nonsmoking white mothers, from both sets (original and censored).

From the censored data (recommended procedure), the analysis of variance (ANOVA) test showed that 3 groups Gr6, Gr7, and Gr9 (out of 6 groups) did not show any significant differences among them at the 99% confidence level. Had we applied the ANOVA test to the original data (inappropriate procedure), the inference would have been that 4 groups Gr6, Gr8, Gr9, and Gr10 do not have significant differences among them. Therefore, it is of utmost importance that the assumption of normality be fulfilled to draw statistically correct inferences from the data at any given confidence level.

Table 1.

Discordant Outlier-Free Major Grouping Statistics of Baby Weights

Group No.	Group Characteristics	Statistical Parameters at Birth (g)						Student t Test at 99% Confidence Level (Inference)
Group No.	Group Characteristics	n	$\bar{x}$	s	se	u ₉₉	CI ₉₉
Gr1	All male babies	468	3417	518	24	62	3355-3479	Male babies have higher weight than female babies.
Gr2	All female babies	473	3225	463	21	55	3170-3280	Male babies have higher weight than female babies.
Gr3	All babies born to non-smoking mothers	831	3334	515	18	46	3288-3380	Babies born to non-smoking mothers have higher weight than those of smoking mothers.
Gr4	All babies born to smoking mothers	122	3162	522	47	124	3038-3285

Notes. n, number of samples or objects (here babies); $\bar{x}$ , sample mean; s, standard deviation; se, standard error of the mean; u₉₉, uncertainty at 99% confidence level; CI₉₉, confidence interval of the mean at 99% confidence level.

Table 2.

Discordant Outlier-Free Major Grouping Statistics of Baby Weights

Group No.	Group Characteristics	Discordancy Tested	Statistical Parameters at Birth						Student t Test at 99% Confidence Level (Inference)
Group No.	Group Characteristics	Discordancy Tested	n	$\bar{x}$	s	se	u ₉₉	CI ₉₉
Gr5	Male baby born to nonsmoking white mother	No	204	3510	602	42	108	3402-3618	The same as below.
Gr6	Male baby born to nonsmoking black mother	No	111	3144	744	71	185	2959-3329	The same as below.
Gr5	Male baby born to nonsmoking white mother	Yes	192	3598	428	31	80	3517-3678	Male babies born to nonsmoking white mothers have higher weight than those born to nonsmoking black mothers.
Gr6	Male baby born to nonsmoking black mother	Yes	101	3319	477	48	125	3194-3444
Gr7	Female baby born to nonsmoking white mother	No	229	3330	538	36	93	3237-3423	The same as below.
Gr8	Female baby born to nonsmoking black mother	No	103	2959	682	67	176	2783-3135	The same as below.
Gr7	Female baby born to nonsmoking white mother	Yes	221	3369	402	27	70	3298-3439	Female babies born to nonsmoking white mothers have higher weight than those of nonsmoking black mothers.
Gr8	Female baby born to nonsmoking black mother	Yes	96	3046	439	45	118	2928-3164
Gr9	Male baby born to smoking white mother	No	52	3211	740	103	279	2932-3490	Male babies born to smoking white mothers do not have more weight than female babies.
Gr10	Female baby born to smoking white mother	No	41	3013	509	79	214	2799-3227
Gr9	Male baby born to smoking white mother	Yes	46	3419	451	66	179	3241-3598	Male babies born to smoking white mothers have more weight than female babies.
Gr10	Female baby born to smoking white mother	Yes	40	3063	398	63	170	2893-3234

Bivariate Data

The errors or uncertainties on any of the individual data (gestational age, mother or father age, and baby weight) are not traditionally reported, eg, gestational age at baby birth traditionally reported cannot be an integer number in weeks; besides, it may not have zero error or uncertainty because the exact date of the last menstruation may not have been known to all mothers. Therefore, we can only evaluate the data from the ordinary least-squares linear regression (OLR) model.

We explored the purely bivariate relationships of gestational age (G_age) and baby weight (W_baby) for different sets of data in Gr5 to Gr10. All regressions from both the original and the censored data are statistically significant at the 99% confidence level (Table 3).

For example, the original complete data for Gr5 “male babies born to nonsmoking white mothers” BiDASys gave the following valid regression equation with statistically significant⁵² linear correlation coefficient r = 0.55561:

W_{b a b y} = - 2055 (\pm 1526) + [143.5 (\pm 39.3) \times G_{a g e}]

(1)

The discordant outlier-free data for Gr5 provided the following regression equation (r = 0.48348):

W_{b a b y} = 513 (\pm 1222) + [77.4 (\pm 31.2) \times G_{a g e}]

(2)

Note the unusual result of lower r value in Equation (2) as compared to Equation (1). The examination of discordant observations showed that they were preferentially removed from the lower gestational age values. Consequently, the regression equation (Equation [2]) lost control on the lower end of the line toward the origin, making the r values lower than the original dataset. This situation also existed with the data of Gr10. In any case, the G_age as the x-variable will lack data from the lowest G_age value to the origin.

For the other 4 groups (Gr6 to Gr9), the bivariate discordancy tests increased the r values (as expected) and reduced the uncertainty of both intercept and slope variables (Table 3). These 4 groups, after separating the discordant bivariate data, showed higher r values (0.74416-0.85317; Table 3) than Gr5 and Gr10.

Although all OLR linear regressions seem to be statistically valid (Pc (r; n) < 0.00005, except one case with Pc (r; n) = 0.0078),^{36, 52} the generally low r and R² values and high uncertainty values on both intercept and slope indicate that more complex regressions, such as multiple linear or polynomial, could provide a better fitting.

Table 3.

Bivariate Ordinary Least-Squares Linear Regressions (OLR; the Grouping Is the Same as in Table 2)

y-Axis Parameter	Discor-Dancy Tested	Statistical Parameters
y-Axis Parameter	Discor-Dancy Tested	Number of Data Pairs	Intercept a	Uncertainty $u_{99}^{a}$	Slope b	Uncertainty $u_{99}^{b}$	Correlation Coefficient r	R²	Probability of No Correlation Pc (r; n)
Gr5: Male baby born to nonsmoking white mother	No	204	–2055	1526	143.5	39.3	0.55561	0.30870	<0.00005
Gr5: Male baby born to nonsmoking white mother	Yes	140	513	1222	77.4	31.2	0.48348*	0.23375	<0.00005
Gr6: Male baby born to nonsmoking black mother	No	111	–2670	1340	152.5	35.1	0.73683	0.54292	<0.00005
Gr6: Male baby born to nonsmoking black mother	Yes	90	–2910	1050	160.2	27.5	0.85317	0.72789	<0.00005
Gr7: Female baby born to nonsmoking white mother	No	229	–2237	1353	143.7	34.9	0.57915	0.33542	<0.00005
Gr7: Female baby born to nonsmoking white mother	Yes	152	–1754	949	130.4	24.3	0.75215	0.56573	<0.00005
Gr8: Female baby born to nonsmoking black mother	No	52	–3710	2639	180	68	0.70564	0.49793	<0.00005
Gr8: Female baby born to nonsmoking black mother	Yes	50	–3475	2340	173	60	0.74416	0.55377	<0.00005
Gr9: Male baby born to smoking white mother	No	54	–4070	2470	189	64	0.73679	0.54286	<0.00005
Gr9: Male baby born to smoking white mother	Yes	52	–3910	2180	184	56	0.77733	0.60424	<0.00005
Gr10: Female baby born to smoking white mother	No	41	–2590	3403	145	88	0.58163	0.33829	<0.00005
Gr10: Female baby born to smoking white mother	Yes	40	–275	3551	86	92	0.38246*	0.14628	0.0078

Notes. *Bivariate discordant outliers removed from the lower end of the gestational age with the consequence that the regression line lost control toward the origin.

Multiple Linear Regression

We used commercial software Statistica^® (Tibco Software, Inc., USA, since May 15, 2017; www.tibco.com/products/tibco-statistica) for processing the data. Unfortunately, the modern high performance single-outlier and recursive discordancy tests are not available to test the normality of the residuals in commercial software for multiple regressions. We evaluated all independent variables (variables: gestational age $G_{age}$ in weeks, mother´s age $M_{age}$ in years, father´s age $F_{age}$ in years, mother´s weight gain $W_{gain}$ in kg, and mother´s visits $M_{vis}$ to the doctor or clinic) to explain the dependent variable of baby weight $W_{baby}$ (in grams).

As an example, we obtained the following equation for Gr5 data (multiple $r$ = 0.59911; $R^{2}$ = 0.35893):

\begin{array}{l} W_{b a b y} = - 2291 (\pm 677) + [136.6 (\pm 15.5) \times G_{a g e}] \\ + [17.6 (\pm 9.8) \times M_{a g e}] + [- 13.9 (\pm 9.4) \times F_{a g e}] \\ + [8.7 (\pm 6.3) \times W_{g a i n}] + [24.1 (\pm 12.3) \times M_{v i s}] \end{array}

(3)

Of the 5 explanatory variables, only 1 ( $G_{age}$ ) was statistically significant at the 99% confidence level (note the lowest relative standard error of the respective coefficient in Equation 3). The quality of multiple linear regression was improved with respect to the bivariate linear regression ( $r$ increased from 0.55561 to 0.59911).

As another example, the multiple linear regression to the data for Gr6 showed that the highest multiple r was obtained from 4 explanatory variables (without $W_{gain}$ ; multiple $r$ = 0.74155; $R^{2}$ = 0.54989):

\begin{array}{l} W_{b a b y} = - 2173 (\pm 626) + [136.9 (\pm 17.7) \times G_{a g e}] \\ + [15.5 (\pm 15.4) \times M_{a g e}] + [- 18.7 (\pm 12.4) \times F_{a g e}] \\ + [19.1 (\pm 17.3) \times M_{v i s}] \end{array}

(4)

In this example also, only 1 variable ( $G_{age}$ ) was statistically significant at the 99% confidence level. The regression quality was improved with respect to the bivariate regression ( $r$ increased from 0.73683 to 0.74155). Other groups (Table 3) can be similarly evaluated.

Multiple Polynomial Regression

As done for the multiple linear regression, polynomial regression could be applied to the data of this case study. As an example, we applied polynomial regression to Gr6 (Table 3) and observed that the highest multiple r was obtained from 4 (not 5) explanatory variables, according to the following equation (multiple $r$ = 0.79145; $R^{2}$ = 0.62640):

\begin{array}{l} W_{b a b y} = - 10861 (\pm 2902) + [606 (\pm 167) \times G_{a g e}] \\ + [- 6.9 (\pm 2.4) \times {(G_{a g e})}^{2}] + [192 (\pm 100) \times M_{a g e}] \\ + [- 2.9 (\pm 1.7) \times {(M_{a g e})}^{2}] + [- 127 (\pm 80) \times F_{a g e}] \\ + [1.6 (\pm 1.2) \times {(F_{a g e})}^{2}] + [35 (\pm 54) \times M_{v i s}] \\ + [- 0.8 (\pm 1.7) \times {(M_{v i s})}^{2}] \end{array}

(5)

Only one variable ( $G_{age}$ ) was statistically significant at the 99% confidence level. The polynomial regression quality was improved with respect to the multiple linear regression (multiple $r$ increased from 0.74155 to 0.79145).

Linear Discriminant Analysis

The complete multivariate data were processed in the linear discriminant analysis (LDA) module of Statistica. Our aim for illustration purposes was to find out if a linear discriminant function could be proposed that will predict if the mother of a singleton baby was a nonsmoker or smoker. We evaluated all 6 explanatory variables ( $W_{baby}$ , $G_{age}$ , $M_{age}$ , $F_{age}$ , $W_{gain}$ , and $M_{vis}$ ) and observed that only 4 of them ( $W_{baby}$ , $G_{age}$ , $M_{age}$ , and $W_{gain}$ ) provided the discrimination at significance (confidence) levels of 0.00002 (99.998%), 0.0405 (95.95%), 0.0204 (97.96%), and 0.0085 (99.15%), respectively. The resulting percent success values were about 61.8% and 63.8%, respectively, for the nonsmoker and smoker mother classes (weighted average of about 62.0%).

Because the LDA assumes multinormality, the 4 variables for individual groups were processed in DOMuDAF to test multinormality and identify multivariate discordant outliers. The discordant outlier-free data were used once again for the LDA and canonical analysis. The overall success increased as compared to the use of the original data (to about 63.4% versus 62.0%). The corresponding discriminant function was as follows:

\begin{array}{l} D F 0 = 3.57877 + [0.00168 \times W_{b a b y}] + [0.06102 \times G_{a g e}] \\ + [- 0.06408 \times M_{a g e}] + [0.09836 \times W_{g a i n}] \end{array}

(6)

Because of the generally low success values in this case study, we will not consider the result for future applications.

The Second Case Study

Original hand measurement data at birth for babies born in a hospital in Lille, France, were reported for 2 groups (Gr11 and Gr12) of estimated gestational ages in weeks (EGA or $G_{age}$ ): (a) Gr11 25 babies of 26 to 36 weeks and (b) Gr12 36 babies of 37 to 41 weeks.⁵³ We applied all techniques from univariate to multivariate for processing and interpreting these data.

Now, because the number of samples in each group was very small (only 25 and 36 in Gr11 and Gr12, respectively; a total of 61), we did not separate the male and female babies for this analysis, although it should always be done when the number of samples is sufficiently large.

From the experimental point of view, we emphasize that the individual hand measurements (7 variables; Table 4) should be repeated several times, preferably by different individuals, and both central tendency and dispersion parameters should be reported. Baby weights should also be determined more than once. Here also, UDASys could be of immense help for obtaining central tendency and dispersion estimates of the initial variables, free from the discordant observations.

Univariate Data

UDASys was used to process the baby characteristics (1 weight and 7 different hand measurements) as univariate data for both sets (before and after of the application of discordancy tests; Table 4). Gr11 did not show any discordant outliers for any of the 8 variables, whereas Gr12 showed them for 6 out of 8 variables (identified by an asterisk in Table 4), for which the mean values changed (decreased or increased) and the dispersion parameters significantly decreased as compared to the original data.

The t test showed for all variables that the babies of lower gestational ages had lower weights and smaller hand sizes (all 7 variables) in all evaluated sample pairs (see the final column in Table 4). The use of discordant outlier-free data did not change the conclusions reached by the original data arrays when the decision was taken at the 99% confidence level (Table 4).

Table 4.

Original and Discordant Outlier-Free Major Grouping Statistics of Baby Characteristics (Gr11: 26 to 36 Weeks EGA or $G_{age}$ and Gr12: 37 to 41 Weeks $G_{age}$ )

Group No.	Group Characteristics (in mm Except when Stated Otherwise)	Discordancy Tested	Statistical Parameters at Birth						Student t Test at 99% Confidence Level (Inference)
Group No.	Group Characteristics (in mm Except when Stated Otherwise)	Discordancy Tested	n	$\bar{x}$	s	se	u₉₉	CI ₉₉
Gr11	W_b: Weight of the baby in g	No	25	1673	590	118	330	1343-2003	The same as below.
Gr12	W_b: Weight of the baby in g	No	36	3241	644	107	292	2949-3534	The same as below.
Gr11	W_b: Weight of the baby in g	Yes	25	1673	590	118	330	1343-2003	The babies of GA of 26-36 weeks have lower weight than those of 37-41 weeks.
Gr12	W_b: Weight of the baby in g*	Yes	29*	3430	345	64	177	3253-3607
Gr11	W_i: Width of the second digit (index)	No	25	6.66	1.14	0.23	0.64	6.02-7.30	The same as below.
Gr12	W_i: Width of the second digit (index)	No	36	8.202	0.931	0.155	0.423	7.779-8.625	The same as below.
Gr11	W_i: Width of the second digit (index)	Yes	25	6.66	1.14	0.23	0.64	6.02-7.30	The babies of GA of 26-36 weeks have smaller index width than those of 37-41 weeks.
Gr12	W_i: Width of the second digit (index)	Yes	36	8.202	0.931	0.155	0.423	7.779-8.625
Gr11	W_t: Width of the first digit (thumb)	No	25	7.41	1.09	0.22	0.61	6.80-8.02	The same as below.
Gr12	W_t: Width of the first digit (thumb)	No	36	9.016	0.931	0.155	0.423	8.594-9.439	The same as below.
Gr11	W_t: Width of the first digit (thumb)	Yes	25	7.41	1.09	0.22	0.61	6.80-8.02	The babies of GA of 26-36 weeks have lower thumb width than those of 37-41 weeks.
Gr12	W_t: Width of the first digit (thumb)	Yes	36	9.016	0.931	0.155	0.423	8.594-9.439
Gr11	R_t: Ray of the first digit (thumb)	No	25	28.82	5.6	1.13	3.16	25.66-31.98	The same as below.
Gr12	R_t: Ray of the first digit (thumb)	No	36	36.57	3.71	0.62	1.68	34.88-38.25	The same as below.
Gr11	R_t: Ray of the first digit (thumb)	Yes	25	28.82	5.6	1.13	3.16	25.66-31.98	The babies of GA of 26-36 weeks have smaller thumb ray than those of 37-41 weeks.
Gr12	R_t: Ray of the first digit (thumb)*	Yes	29*	36.45	1.95	0.36	1.0	35.45-37.45
Gr11	L_{_m}: Length of the middle digit	No	25	23.56	2.56	0.51	1.43	22.13-25.0	The same as below.
Gr12	L_{_m}: Length of the middle digit	No	36	27.98	2.58	0.43	1.17	26.81-29.15	The same as below.
Gr11	L_{_m}: Length of the middle digit	Yes	25	23.56	2.56	0.51	1.43	22.13-25.0	The babies of GA of 26-36 weeks have smaller middle digit length than those of 37-41 weeks.
Gr12	L_m: Length of the middle digit*	Yes	34*	28.43	1.81	0.31	0.85	27.58-29.28
Gr11	L_p: Length of the palm of the hand	No	25	27.30	4.31	0.86	2.41	24.89-29.71	The same as below.
Gr12	L_p: Length of the palm of the hand	No	36	34.10	3.45	0.58	1.57	32.53-35.67	The same as below.
Gr11	L_p: Length of the palm of the hand	Yes	25	27.30	4.31	0.86	2.41	24.89-29.71	The babies of GA of 26-36 weeks have smaller palm length than those of 37-41 weeks.
Gr12	L_p: Length of the palm of the hand*	Yes	29*	35.48	1.83	0.34	0.94	34.53-36.42
Gr11	L_h: Maximal length of the hand	No	25	50.40	5.93	1.19	3.32	47.08-53.71	The same as below.
Gr12	L_h: Maximal length of the hand	No	36	62.01	5.26	0.88	2.40	59.61-64.41	The same as below.
Gr11	L_h: Maximal length of the hand	Yes	25	50.40	5.93	1.19	3.32	47.08-53.71	The babies of GA of 26-36 weeks have smaller maximal hand length than those of 37-41 weeks.
Gr12	L_h: Maximal length of the hand*	Yes	34*	62.93	3.64	0.62	1.7	61.23-64.63
Gr11	W_h: Width of the hand	No	25	26.23	3.46	0.69	1.94	24.29-28.17	The same as below.
Gr12	W_h: Width of the hand	No	36	33.69	3.00	0.50	1.36	32.33-35.05	The same as below.
Gr11	W_h: Width of the hand	Yes	25	26.23	3.46	0.69	1.94	24.29-28.17	The babies of GA of 26-36 weeks have smaller hand width than those of 37-41 weeks.
Gr12	W_h: Width of the hand*	Yes	34*	34.19	2.19	0.38	1.03	33.17-35.22

Notes. Gr11: EGA 26-36 weeks; Gr12: EGA 26-36 weeks; W_i: width of the second digit (index) measured at the mid phalanx, Just above the proximal interphalangeal joint; W_t: width of the first digit (thumb) measured at the middle of the proximal phalanx; R_t: ray of the first digit (thumb) measured from the proximal end of the hand palm to the distal end of the thumb; L_{_m}: length of the middle digit, measured from the base of the digit; L_p: length of the palm of the hand, measured from the proximal end of the hand to the distal end of the middle finger; L_h: maximal length of the hand, measured from the proximal end of the hand to the distal end of the middle digit; W_h: width of the hand, measured on the palm, just below the joint between the metacarpals and the proximal phalanges; * identifies the cases for which univariate discordant outliers were detected and separated.

Table 5.

Bivariate Ordinary Least-Squares Linear Regressions of Different Parameters for all Babies at Birth Against their Gestational Age (Table 4)

y-Axis Parameter	Discor-Dancy Tested	Statistical Parameters
y-Axis Parameter	Discor-Dancy Tested	Number of Data Pairs n	Intercept a	Uncertainty $u_{99}^{b}$	Slope b	Uncertainty $u_{99}^{b}$	Correlation Coefficient r	R ²	Probability of No Correlation Pc (r; n)
W_b: Baby weight (g)	No	61	–5319	1513	217.7	41.3	0.87693	0.76901	<0.00005
W_b: Baby weight (g)	Yes	51*	–5052	930	211.4	25.6	0.95341	0.90899	<0.00005
W_i: Index width (mm)	No	61	–0.21	2.97	0.214	0.081	0.67384	0.45406	<0.00005
W_i: Index width (mm)	Yes	51*	–1.07	2.23	0.233	0.060	0.82745	0.68467	<0.00005
W_t: Thumb width (mm)	No	61	–0.29	2.67	0.238	0.073	0.74852	0.56029	<0.00005
W_t: Thumb width (mm)	Yes	53*	–1.12	2.20	0.262	0.060	0.85338	0.72826	<0.00005
R_t: Thumb ray (mm)	No	61	–9.9	11.4	1.189	0.311	0.79812	0.63700	<0.00005
R_t: Thumb ray (mm)	Yes	50*	–5.5	8.9	1.076	0.242	0.86515	0.74849	<0.00005
L_m: Middle digit length (mm)	No	61	2.7	6.8	0.645	0.186	0.76829	0.59027	<0.00005
L_m: Middle digit length (mm)	Yes	57*	3.9	5.6	0.618	0.152	0.82631	0.68279	<0.00005
L_p: Palm length (mm)	No	61	–5.9	1.022	0.098	0.260	0.80595	0.64955	<0.00005
L_p: Palm length (mm)	Yes	39*	–4.8	1.007	0.048	0.129	0.96121	0.92392	<0.00005
L_h: Maximal hand length (mm)	No	61	–5.4	12.6	1.723	0.344	0.86622	0.75034	<0.00005
L_h: Maximal hand length (mm)	Yes	43*	–5.8	5.2	1.747	0.141	0.98213	0.96458	<0.00005
W_h: Hand width (mm)	No	61	–8.0	7.5	1.063	0.205	0.87354	0.76307	<0.00005
W_h: Hand width (mm)	Yes	47*	–8.7	4.8	1.079	0.130	0.95745	0.91670	<0.00005

Note. * Identifies the cases for which bivariate discordant outliers were detected and separated.

Bivariate Data

The OLR model was used to evaluate bivariate relationships of the 8 variables (1 weight and 7 hand measurements) with $G_{age}$ (weeks). BiDASys provided regression results for both the original data (61 pairs) and bivariate discordant outlier-free data (39 pairs for the palm length to 57 pairs for the middle digit length). Both sets of regressions (Table 5) showed statistically significant correlation at the 99.5% confidence level. The linear correlation coefficient (r) increased from the range of 0.67384-0.87693 to 0.82631-0.98213, and correspondingly, the uncertainty on both intercept and slope of the regression line decreased (Table 5).

Multiple Linear Regression

If we were to explore the relationship (or dependence) of baby weight with other continuous variables, this case study is not useful, because only one explanatory variable ( $G_{age}$ ) was available to explain the baby weight.

Polynomial Regression

As an example, the linear and polynomial regressions (Equations [7] and [8]) for the baby weight ( $W_{b}$ ; y-axis) versus gestational age ( $G_{age}$ ; x-axis) were as follows:

W_{b} = - 5319 (\pm 1513) + [217.7 (\pm 41.3) \times G_{a g e}]

(7)

\begin{array}{l} W_{b} = 7866 (\pm 4705) + [- 554 (\pm 274) \times G_{a g e}] \\ + [11.1 (\pm 3.9) \times {(G_{a g e})}^{2}] \end{array}

(8)

The inclusion of a square term increased the $R^{2}$ to 0.79687 (r = 0.89268), with respect to the linear fit ( $R^{2}$ = 0.76901; r = 0.87693; Table 5). Higher order polynomial(s) may improve the fit, but it is more likely that additional explanatory variables, such as mother’s age or mother’s weight gain, may do the job better.

Linear Discriminant Analysis

The LDA and canonical analysis were applied through Statistica to the complete dataset with the $G_{age}$ as the grouping or class variable, dividing them into 2 groups⁵³ and the different hand measurements and baby weight as independent or explanatory variables. We aimed to obtain a discriminant function from this training set that can be used to predict the group to which a new baby’s mother (baby outside the training set) would belong to. Preliminary analysis of 8 variables (W_b, W_i, W_t, R_t, L_m, L_p, L_h, and W_h; Table 5) showed that 2 of them (L_p and L_h) did not contribute to the improvement of the discrimination.

From 6 explanatory variables (W_b, W_i, W_t, R_t, L_m, and W_h), the percent success values (correct classification) for the 2 groups Gr11 and Gr12 (Table 4) were 96% (24 correctly classified out of 25) for the first group (Gr11) and 86% (31 correctly classified out of 36) for the second group (Gr12). Because the individual groups should be multinormally distributed in the space of these 6 variables (W_b, W_i, W_t, R_t, L_m, and W_h), DOMuDAF was used to evaluate multinormality. The results showed that the complete dataset complied with multinormality, ie, there were no discordant outliers. Therefore, the above-mentioned results are statistically valid. The canonical analysis provided the following discriminant function $DF 0$ equation:

\begin{array}{l} D F 0 = (0.001992 \times W_{b}) + (0.202046 \times W_{i}) \\ + (- 0.255280 \times W_{t}) + (- 0.038320 \times R_{t}) \\ + (- 0.242936 \times L_{m}) + (0.144694 \times W_{h}) \\ + (- 1.365166) \end{array}

(9)

The discriminant scores ( $DF 0$ ) were plotted in Figure 1 for the complete dataset.⁵³ This one-axis diagram (Figure 1) can be used to deduce the group to which a new sample would belong, ie, if the mother was a nonsmoker or smoker. In fact, the probability for the mother belonging to these 2 categories could be calculated.⁵⁴

Nevertheless, we clarify that we have only traced the methodology through which statistical interpretation and inferences could be achieved. The results would be much more useful if the database were representative of the problem at hand.

Discussion and Conclusion

It is generally believed that robust parameters are superior to the outlier-based methods, because the former can cope with or are immune to the presence of outlying observations.⁵⁵ However, recent precise and accurate experiment-compatible Monte Carlo simulations have shown that all robust parameters are affected by asymmetrical contamination, and outlier-based parameters, when correctly used from the application of appropriate discordancy tests (combination of single-outlier and recursive tests), and may provide better results (closer to the “truth,” which is fortunately known from the Monte Carlo simulations) than the robust parameters.^{29, 34, 35, 38-40, 49, 54}

For the data summarized in Table 2, the mean values increased by about 1.2% to 6.5%, whereas the standard deviation and uncertainty values decreased by about 22% to 39% and 21% to 36%, respectively. Similarly, for the data of Table 4, the mean values changed from –0.3% to +5.8%, whereas the standard deviation and uncertainty decreased by about 27% to 47% and 24% to 40%, respectively.

Therefore, the use of mean and standard deviation (or uncertainty) parameters to express the central tendency and dispersion estimates is recommended only after the application of appropriate discordancy tests. Appropriate freely available computer programs (UDASys2³⁴ and UDASys3³⁵) were mentioned and used to illustrate the statistical methods.

For handling of the bivariate data, only the OLR could be used, although 2 types of regressions (OLR and UWLR) were mentioned in this article (see also BiDASys³⁶). For the UWLR, errors or uncertainty values are required for each bivariate data pair, which are not reported in any such medical study. However, the OLR method also assumes that the errors are only on the y-axis, the x-axis is error-free, and all errors are homoscedastic.³⁶ It is likely that none of these assumptions is fulfilled in the medical data interpreted in this article. Therefore, an attempt should be made to estimate errors on individual datum, which will enable us to eventually apply the recommended UWLR model.^{36, 54} Furthermore, this procedure will also allow us to design better experiments to reduce the errors and thus obtain valid inferences in future.

The multivariate data can be handled for multiple linear and polynomial regressions through commercial software, but the multivariate discordant outliers cannot be evaluated from modern discordancy tests in such software. Therefore, appropriate software for doing the combined job (multiple regressions and discordancy of residuals) is still required to be developed. Nevertheless, most evaluations showed that only the gestational age was statistically significant to explain the baby weight, although the use of all variables provided higher R² value than the bivariate regression.

The LDA and canonical analysis constitute a powerful supervised multivariate statistical technique to put forth discriminant functions that can be used for future decisions or applications. Appropriate multivariate data are required on representative samples. In the first case study, the quality of individual data is not precisely known. Nevertheless, it is likely that the poor quality and associated high errors or uncertainties of these variables probably resulted in low success values, although the randomly chosen samples may certainly represent the human population of North Carolina.

The high success values provided in the second case study are interesting, but the small sample sizes are probably not representative of the population, not even in a single hospital in France. The second study, on the other hand, presented better controlled hand measurement experimental variables appropriate for the LDA and canonical analysis. Finally, we also stress that the multinormality assumption inherent to the LDA, and that canonical analysis can be easily fulfilled from the freely available computer program DOMuDAF.³⁷

Footnotes

Acknowledgements

This work was developed when the second author had a summer internship in Mexico at the Instituto de Energías Renovables of the Universidad Nacional Autónoma de México. She is thankful to her family for providing support for her stay in Mexico.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Surendra P. Verma

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Statistical Methodology for Medical Data: Gestational Age,Birth Weight,and Hand Characteristics

Abstract

Abstract

Keywords

Introduction

Recent Advances in Statistical Handling of Data

The First Case Study

Univariate Data

Discordant Outlier-Free Major Grouping Statistics of Baby Weights

Discordant Outlier-Free Major Grouping Statistics of Baby Weights

Bivariate Data

Bivariate Ordinary Least-Squares Linear Regressions (OLR; the Grouping Is the Same as in Table 2)

Multiple Linear Regression

Multiple Polynomial Regression

Linear Discriminant Analysis

The Second Case Study

Univariate Data

Original and Discordant Outlier-Free Major Grouping Statistics of Baby Characteristics (Gr11: 26 to 36 Weeks EGA or G age and Gr12: 37 to 41 Weeks G age )

Bivariate Ordinary Least-Squares Linear Regressions of Different Parameters for all Babies at Birth Against their Gestational Age (Table 4)

Bivariate Data

Multiple Linear Regression

Polynomial Regression

Linear Discriminant Analysis

Discussion and Conclusion

Footnotes

Acknowledgements

Declaration of Conflicting Interests

Funding

ORCID iD

References

Original and Discordant Outlier-Free Major Grouping Statistics of Baby Characteristics (Gr11: 26 to 36 Weeks EGA or $G_{age}$ and Gr12: 37 to 41 Weeks $G_{age}$ )