Testing and Visualization of Associations in Three-Way Contingency Tables: A Study of the Gender Gap in Patients with Type 1 Diabetes and Cardiovascular Complications

Abstract

Using data from the Swedish National Diabetes Register, this study examines the gender disparity among patients with type 1 diabetes who have experienced a specific cardiovascular complication, while exploring the association between their weight variability, age group, and gender. Fourteen cardiovascular complications have been considered. This analysis is conducted using three-way correspondence analysis (CA), which allows for the partitioning and decomposition of Pearson’s three-way chi-squared statistic. The dataset comprises information organized in a data cube, detailing how weight variability among these patients correlates with a cardiovascular complication, age group, and gender. The three-way CA method presented in this paper allows one to assess the statistical significance of the association between these variables and to visualize this association, highlighting the gender gap among these patients. From this analysis, we find that the association between weight variability, age group, and gender varies among different types of cardiovascular complications.

1. Introduction

Patients diagnosed with type 1 diabetes (T1D) face a heightened risk of developing a range of cardiovascular complications (CVCs) compared to the general population [1,2]. It is known that women with T1D face a greater risk of all-cause mortality and experience more vascular events compared to men [3]. This study assesses gender differences in patients with T1D [4,5,6], focusing on weight variability and age in males and females who experienced a CVC, and examining the associations between these variables in different cohorts.

By utilizing data from the Swedish National Diabetes Register (NDR), this paper explores the links between weight variability, age group, and gender of patients with T1D and a CVC. The CVCs that we focus our attention on are: (1) stroke, (2) myocardial infarction (myo), (3) heart failure hospitalization (HFH), (4) coronary artery bypass graft (CABG), (5) percutaneous coronary intervention (PCI), (6) peripheral artery disease (PAD), (7) renal failure (RF), (8) acute renal disease (ARF), (9) chronic renal failure (CRF), (10) dialysis, (11) glomerular filtration rate of 50 mL/min/1.73 m² (GFR50), (12) macroalbuminuria (Macr), (13) retinopathy (Ret), and (14) all-cause mortality (Mort).

This population-based cohort study involved 6255 participants (3485 males and 2770 females) with T1D, ranging in age from 21 to 85 years, identified within the NDR. The inclusion period began on 1 July 2003 and concluded on 30 December 2013, during which time participants underwent at least three distinct weight measurements. Weight variability was assessed by computing the coefficient of variability (CV) for weight measurements of each patient and dividing it into quartiles. Subsequently, data concerning each of the 14 CVCs were summarized into a three-way contingency table, stratified by weight variability quartiles, age quartiles, and gender. To study the dependence relationship between the patients’ weight variability, age group and gender, we consider a three-way correspondence analysis (CA) [7,8] based on the partition and decomposition of a three-way index of association, that is, Pearson’s chi-squared statistic [9,10]. Pearson’s three-way chi-squared statistic is orthogonally partitioned into two-way and three-way association terms to determine the dependence relationships between variables. Note that we use the term “association” when analyzing the dependence between variables and the term “interaction” when analyzing the dependence between the categories of the variables. Indeed, three-way CA allows us to decompose the association between the variables and also to visualize the interactions between the diverse categories.

The paper is organized as follows. After introducing the notation to be used in this paper in Section 2, Section 3 and Section 4 provide a brief description of the partition of Pearson’s three-way chi-squared statistic and of the classical three-way CA technique, respectively. Section 5 gives a brief overview of the use and construction of biplots for three-way correspondence analysis. Section 6 describes several illustrative examples demonstrating how the association between weight variability, age group, and gender varies with respect to a given CVC. Section 7 provides some final remarks.

2. Materials and Methods

Before analyzing the data, it is necessary to explain some of the general principles behind three-way CA.

Notation and Terminology

Define an $I\times J\times K$ (three-way) contingency table by $\underline{\mathbf{N}}$, which contains information on the three cross-classified categorical variables observed on n individuals. This contingency table consists of I rows, J columns, and K tubes, and its ($i,j,k$)th element is denoted by $n_{ijk}$. Let $p_{ijk}=n_{ijk}/n$ be the proportion of observations in cell $(i,j,k)$, and $p_{i••}$, $p_{•j•}$, and $p_{••k}$ be the $i\mathrm{th}$ row, $j\mathrm{th}$ column, and $k\mathrm{th}$ tube marginal proportions, respectively, also called expected proportions. The deviations from the three-way independence model are defined as follows:

$$\begin{array}{c}\hfill {\pi }_{ijk}=\frac{p_{ijk}}{p_{i••}{p}_{•j•}{p}_{••k}}-1.\end{array}$$

The value ${\pi }_{ijk}$ will primarily be referred to as Pearson’s residual for the cell $(i,j,k)$, and its square is a cell’s contribution to the chi-squared statistic $X_{IJK}^2$. We define ${\mathbf{D}}_I$ to be the $I\times I$ diagonal matrix with the row marginal $p_{i••}$ as its $\left(i,i\right)$th element. Similarly, ${\mathbf{D}}_J$ and ${\mathbf{D}}_K$ are the $J\times J$ and $K\times K$ diagonal matrices of the column and tube marginal proportions, respectively. Finally, we consider the total inertia or phi-squared statistic, ${\Phi }^2=X_{IJK}^2/n$, which is also equal to the squared norm of the table $\underline{\mathbf{\Pi }}=\left({\pi }_{ijk}\right)$ containing the Pearson residuals so that

$$\begin{array}{c}\hfill {\Phi }^2=\frac{X_{IJK}^2}{n}=\left|\right|\underline{\mathbf{\Pi }}{\left|\right|}^2=\sum _{i=1}^I\sum _{j=1}^J\sum _{k=1}^K{p}_{i••}{p}_{•j•}{p}_{••k}{\left(\frac{p_{ijk}}{p_{i••}{p}_{•j•}{p}_{••k}}-1\right)}^2.\end{array}$$

Using ${\Phi }^2$, Carlier and Kroonenberg [7] proposed a three-way CA method employing the Tucker3 model [11], which is the most common generalization of the singular value decomposition of a matrix; see also Kroonenberg (p. 54, [12]). This model defines components for each of the three categorical variables. These components are a linear combination of the original variables, synthesizing the information from the categories of variables. It also consists of a three-way core table containing the three-way equivalent of singular values; see also Lombardo, Beh, and Kroonenberg [8]; Lombardo, van de Velden, and Beh [13]; and Beh and Lombardo (Chapter 11, [14]). The elements of the core table reflect the strengths of the links between the components of the three variables. The decomposition does not impose any restrictions on the components other than the identification constraints of orthogonality.

3. Partitioning the Chi-Squared Statistic

Lombardo, Takane, and Beh [15] discuss a common method for studying the association among three categorical variables based on the orthogonal decomposition of Pearson’s chi-squared statistic, $X_{IJK}^2$, and under a variety of hypotheses about the expected proportions (hypothesized probabilities). These partitions are based on the ANOVA-like family-wise partitions of Pearson’s three-way chi-squared statistic. In this paper, we focus on the most common case where these probabilities are estimated using the marginal proportions of the empirical distribution underlying the data. In some cases, however, the probabilities are prescribed by the analyst, because a priori knowledge of the phenomena suggests otherwise [16].

Under the independence assumption, Pearson’s three-way chi-squared statistic can be partitioned into four orthogonal parts, each corresponding to an association term, as detailed, for example, by Lancaster [17]. This partition of $X_{IJK}^2$ is given by

$$\begin{array}{c}\hfill X_{IJK}^2=X_{IJ}^2+X_{IK}^2+X_{JK}^2+X_{int}^2\end{array}$$

(1)

and can be written in more detail as

$$\begin{array}{ccc}\hfill X_{IJK}^2& =& n\sum _{i=1}^I\sum _{j=1}^J{p}_{i••}{p}_{•j•}{\left(\frac{p_{ij•}-p_{i••}{p}_{•j•}}{p_{i••}{p}_{•j•}}\right)}^2+n\sum _{i=1}^I\sum _{k=1}^K{p}_{i••}{p}_{••k}{\left(\frac{p_{i•k}-p_{i••}{p}_{••k}}{p_{i••}{p}_{••k}}\right)}^2\hfill \\ & +& n\sum _{j=1}^J\sum _{k=1}^K{p}_{•j•}{p}_{••k}{\left(\frac{p_{•jk}-p_{•j•}{p}_{••k}}{p_{•j•}{p}_{••k}}\right)}^2+n\sum _{i=1}^I\sum _{j=1}^J\sum _{k=1}^K{p}_{i••}{p}_{•j•}{p}_{••k}{\left(\frac{p_{ijk}{-}_{\alpha }{p}_{ijk}}{p_{i••}{p}_{•j•}{p}_{••k}}\right)}^2,\hfill \end{array}$$

(2)

where

$$\begin{array}{c}\hfill {\hspace{0.17em}}_{\alpha }{p}_{ijk}=p_{ij•}{p}_{••k}+p_{i•k}{p}_{•j•}+p_{•jk}{p}_{i••}-2p_{i••}{p}_{•j•}{p}_{••k}.\end{array}$$

The first three terms of Equations (1) and (2) are the two-way association terms obtained by a weighted aggregation of the total association measure across each of the variables. For example, $X_{IJ}^2$ is Pearson’s chi-squared statistic for the row and column variables formed by aggregating across each of the tube categories. The last term, $X_{int}^2$, represents the trivariate association between all three categorical variables. For further details on this partition, see Carlier and Kroonenberg (p. 358, [7]); Beh and Lombardo (p. 454, [14]); and Lombardo, Takane, and Beh [15].

The $X_{IJK}^2$, as well as the four association terms on the right-hand side of Equations (1) and (2), can be tested for (asymptotic) statistical significance since all these terms are asymptotic chi-squared random variables. Specifically, under the null hypotheses that all three variables are independent, the statistical significance of $X_{IJK}^2$ can be assessed using $\left(I-1\right)\left(J-1\right)+\left(I-1\right)\left(K-1\right)+\left(J-1\right)\left(K-1\right)+\left(I-1\right)\left(J-1\right)\left(K-1\right)$ degrees of freedom. The statistical significance of the first term, $X_{IJ}^2$, can be evaluated using $\left(I-1\right)\left(J-1\right)$ degrees of freedom (df) and the second term, $X_{IK}^2$, by using $\left(I-1\right)\left(K-1\right)$ df. Similarly, the statistical significance of the third term, $X_{JK}^2$, can be assessed using $\left(J-1\right)\left(K-1\right)$ df and the fourth term, $X_{IJK}^2$, by using $\left(I-1\right)\left(J-1\right)\left(K-1\right)$ df.

4. Modeling Association

When the analyst not only wishes to test the statistical significance of the three-way association but also wants to provide a graphical representation of it, a decomposition of the three-way contingency table is necessary for exploring in more detail the interactions between the categories.

The Tucker3 Decomposition

As indicated in Section 2, the most common three-way generalization of the singular value decomposition within the context of three-way CA is the Tucker3 model proposed by Tucker [11]. The general form of the decomposition is

$$\begin{array}{c}\hfill {\pi }_{ijk}={\widehat{\pi }}_{ijk}+e_{ijk}=\sum _{p=1}^P\sum _{q=1}^Q\sum _{r=1}^R{g}_{pqr}{a}_{ip}{b}_{jq}{c}_{kr}+e_{ijk},\end{array}$$

(3)

where P, Q, and R (with $P\le I,Q\le J$, and $R\le K$) represent the number of columns in the component matrices $\mathbf{A},\mathbf{B}$, and $\mathbf{C}$ that are associated with the rows, columns, and tube categories, respectively. The sets of component vectors ${\mathbf{a}}_p\left\{a_{ip}\right\},{\mathbf{b}}_q\left\{b_{jq}\right\},$ and ${\mathbf{c}}_r\left\{c_{kr}\right\}$ are orthonormal with respect to ${\mathbf{D}}_I$, ${\mathbf{D}}_J$, and ${\mathbf{D}}_K$, respectively. The quantities $g_{pqr}$ are the elements of the core table and can be interpreted as the generalized or three-way analogs, of the two-way singular values, where its magnitude indicates the strength of the relationship among the components. The magnitude of the squared core element, $g_{pqr}^2$, also indicates the contribution of the product of the P, Q, and R components to the fit of the model to the data, allowing the analyst to compute the inertia explained in a two-dimensional plot. Note that, given the orthogonality of the components, the phi-squared statistic or total inertia can be expressed as the sum of squares of the elements of the core table, so that

$$\frac{X_{IJK}^2}{n}=\sum _{p=1}^I\sum _{q=1}^J\sum _{r=1}^K{g}_{pqr}^2.$$

(4)

For more details on Equation (4), see Kroonenberg (pp. 226–227, [12]).

Finally, the element $e_{ijk}$ in Equation (3) is the error of approximation between the observed ${\pi }_{ijk}$ and their predicted values, ${\widehat{\pi }}_{ijk}$; see, for example, Kroonenberg (Chapter 4, [12]. The choice of the number of row, column, and tube components, P, Q, and R, respectively, relies on various methodologies developed by researchers for the Tucker3 model. Studies such as Kiers [18,19], Ten Berg and Kiers [20], Timmerman and Kiers [21], Rocci [22], Kroonenberg [23], and Lombardo, van de Velden, and Beh [13] illustrate this point. Here, we will adopt the strategy of Lombardo, van de Velden, and Beh [13] to choose P, Q, and R. This strategy uses the tunelocal() function in the R package CA3variants (v. 3.3.1). The three-way CA performed in Section 6 uses the CA3variants() function of this package.

5. Biplots in Three-Way Correspondence Analysis

To visualize the association structure among two or more categorical variables, we limit ourselves to the biplot, a technique first introduced by Gabriel [24]. Comprehensive discussions of the variety of biplots that can be constructed, particularly in the context of CA, are available in the works of Carlier and Kroonenberg [7]; Lombardo, Carlier, and D’Ambra [25]; Greenacre [26]; and Beh and Lombardo [27].

Biplots for Three-Way Correspondence Analysis

To illustrate the association structure among the categorical variables of a three-way contingency table, Carlier and Kroonenberg [7,28] explored different biplot techniques. The technique of interest here employs a single component for one variable and the interactively-coded components for the other variables; this biplot is also known as a nested-mode biplot (Sect. 11.5.4, [12]) or interactive biplot (p. 472, [14]). Lombardo, van de Velden, and Beh [13] discuss six variants of the interactive biplot. These variants depend on the variables chosen for the interactive representation and on the computation of the type of coordinates, i.e., standard or principal coordinates [29,30]. Firstly, we can represent the coordinates of the row categories using their $\mathbf{A}$ components, while the coordinates of the column and tube categories are represented interactively by the Kronecker product of their $\mathbf{B}$ and $\mathbf{C}$ components. Secondly, the coordinates of the column categories can be represented by their $\mathbf{B}$ components, while the coordinates of the rows and tubes are interactively represented by the Kronecker product of their $\mathbf{A}$ and $\mathbf{C}$ components. Thirdly, the coordinates of the tube categories can be represented by their $\mathbf{C}$ components, with the coordinates of the rows and columns being interactively represented by the Kronecker product of their $\mathbf{A}$ and $\mathbf{B}$ components.

When the coordinates are multiplied by the core elements, they are referred to as principal coordinates; otherwise, they are standard coordinates. Their features are the same as those derived for biplots in the classical approach to CA [30]; for more information, refer to Greenacre [26] and Beh and Lombardo (Chapter 11, [14]). In particular, the distances between the categories from the same variable are correctly portrayed only when using the principal coordinates.

The interactive coordinates can be expressed in either their standard or principal coordinate form. Therefore, the three previous types of variable representation lead to six types of interactive biplots; for further details, see Lombardo, van de Velden, and Beh (pp. 243–244, [13]).

6. Practical Application: Patients with T1D and CVCs

6.1. Preliminary Discussion of the NDR Data

We utilize data from the NDR to assess and visualize the association between the variables weight variability, age group, and gender among patients with T1D who experienced one of the 14 CVCs outlined in Section 1. We classify their weight variability using four quartiles: the first quartile includes patients with a CV of weight between 0 and 0.018 (Q1: low); the second quartile includes those with a CV of weight between 0.018 and 0.028 (Q2: moderately low); the third quartile includes those with a CV of weight between 0.028 and 0.043 (Q3: middle-high); and the fourth quartile includes those with a CV of weight at least 0.043 (Q4: high). The CV of the weight variable is then cross-classified to create a three-way contingency table with respect to the four age group quartiles (age1 = 21–27 years; age2 = 27–40 years; age3 = 40–52 years; age4 = 52+ years) and to gender (males and females). Consequently, the data are summarized in 14 three-way contingency tables, with each table corresponding to a specific CVC.

Table 1. Test of independence between weight variability, age group, and gender for each of the 14 CVCs using Pearson’s three-way chi-squared statistic with 24 df.

CVC	${\mathit{X}}_{\mathit{IJK}}^2$	p-Value
Stroke	26.051	0.351
Myocardial infarction	39.932	0.022
HFC	33.788	0.089
CABG	26.695	0.319
PCI	23.067	0.516
PAD	29.429	0.204
Renal failure	29.536	0.201
Acute renal failure	30.685	0.163
Chronic renal failure	28.312	0.247
Dialysis	14.212	0.942
GFR50	32.723	0.110
Macroalbuminuria	73.806	<0.001
Retinopathy	168.271	<0.001
All-cause mortality	38.280	0.032

Those p-values in bold are less than the nominal significance level of 0.01.

presents Pearson’s three-way chi-squared statistic for each of the 14 CVCs given in Section 1, as well as their p-value with respect to 24 df. Interestingly, Pearson’s three-way chi-squared statistic does not yield statistically significant results for all 14 CVCs. Rather, it only shows that there is a statistically significant association (at the 1% level of significance) for two CVCs: macroalbuminuria and retinopathy. Therefore, among the 14 CVCs, there is no statistically significant association between weight variability, age group, and gender for any of the six cardiovascular diseases (stroke, myo, HFH, CABG, PCI, and PAD). The interested reader can find the related three-way tables in the Supplementary Materials for this paper.

In the following sections, we discuss the partition and decomposition of Pearson’s chi-squared statistic for the CVCs macroalbuminuria and retinopathy. We focus exclusively on these two three-way contingency tables, since it is only for these CVCs where a statistically significant association exists between weight variability, age group, and gender.

6.2. Patients with TD1 and Macroalbuminuria

The main question we shall address here is:

To what extent is there an association between weight variability, age group, and gender for patients with T1D and macroalbuminuria?

Table 2. Patients with T1D and macroalbuminuria, cross-classified by weight variability, age group, and gender.

Gender: Male
Age Group
Weight Variability	Age1	Age2	Age3	Age4
Q1	7	14	24	38
Q2	12	23	18	24
Q3	17	19	23	12
Q4	18	15	11	6
Gender: Female
Age Group
Weight Variability	Age1	Age2	Age3	Age4
Q1	7	10	11	20
Q2	14	12	7	10
Q3	11	15	18	8
Q4	21	10	13	5

presents a cross-classification of weight variability (row variable), age group (column variable), and gender (tube variable) among patients with T1D experiencing macroalbuminuria.

To assess the nature of the association between weight variability, age group, and gender in detail, we partition Pearson’s three-way chi-squared statistic, $X_{IJK}^2$, first in accordance with Equation (1) so that three bivariate terms and a trivariate term are produced.

Table 3. Partition of Pearson’s three-way association statistic for patients with T1D and macroalbuminuria in Table 2.

	$\mathit{Term}-\mathit{IJ}$	$\mathit{Term}-\mathit{IK}$	$\mathit{Term}-\mathit{JK}$	$\mathit{Term}-\mathit{IJK}$	$\mathit{Term}-\mathit{Total}$
chi-squared	57.959	5.373	5.294	5.180	73.806
phi-squared	0.123	0.011	0.011	0.011	0.156
% of Inertia	78.528	7.281	7.173	7.018	100.000
df	9	3	3	9	24
p-value	<0.001	0.146	0.151	0.818	<0.001
Term/df	6.440	1.791	1.765	0.576	3.075

Those p-values in bold are less than the nominal significance level of 0.01.

summarizes the partition of this statistic for Table 2. This includes the partition of phi-squared into four terms: their percentage contribution to phi-squared, the corresponding degrees of freedom (df), the p-value, and the relative size ($Term$/df) of each partition term (which facilitates a comparison of the chi-squared values derived from different asymptotic chi-squared distributions). In Table 3, looking at Pearson’s three-way statistic $X_{IJK}^2=73.806$ and its p-value, which is less than $0.001$ (df = 24), we can conclude that there is clear evidence of a statistically significant association among the three variables of Table 2. Table 3 also shows that the most dominant source of association among the three variables is the association between weight variability and age group, denoted as $Term-IJ$; it contributes to over 75% of $X_{IJK}^2$ and has a p-value that is less than 0.001. While the p-value of $X_{IJK}^2$ for Table 1 shows there is a statistically significant association between the variables, Table 3 reveals that not all three bivariate terms of the partition yield a statistically significant association between the variables. Specifically, only $Term-IJ$ is statistically significant. Conversely, the associations between weight variability and gender ($Term-IK$) and between age group and gender ($Term-JK$) are not statistically significant, nor is the trivariate term ($Term-IJK$). This means that, while not all sources of associations between the variables are statistically significant, at least one is. Also, while $Term-IJK$ is not statistically significant, we can still identify categories and joint categories that do provide a statistically significant contribution to the overall association among the three variables. We now turn our attention to addressing this issue for Table 2.

To identify which categories of Table 2 provide a statistically significant contribution to the total association, we calculate the p-value for each quartile of the weight variability variable. We also calculate the p-value of the interaction between the age group and gender variables. These p-values are determined using the approach described in Beh and Lombardo [31]. Their methodology constructs 100$(1-\alpha )\%$ confidence ellipses for points in a low-dimensional plot generated from applying a CA to the data. The methodology also includes ways to calculate excellent p-value approximations for each point when assessing their contribution to the association (or lack thereof) between categorical variables. These p-values are designed to reflect the statistical significance of the distance of a category’s point from the origin, thus determining the statistical significance of the category to the association structure between the categorical variables. They are calculated taking into account the chi-squared statistic, the principal coordinates, the inertia explained by each axis, and the marginal proportions of $\underline{\mathbf{N}}$.

Table 4. The p-values of the categories of weight variability and age group for males and females with T1D and macroalbuminuria.

p-values of Row Principal Coordinates
Category	p-value
Q1	0.003
Q2	0.027
Q3	0.001
Q4	0.001
p-values of Column–Tube Principal Coordinates
Category	p-value
age1Male	0.957
age2Male	0.998
age3Male	0.934
age4Male	0.047
age1Female	0.005
age2Female	1.000
age3Female	0.158
age4Female	0.959

Those p-values in bold are less than the nominal significance level of 0.01.

shows that all the categories of weight variability are statistically significant at the $\alpha =0.01$ level, except for Q2. However, only two categories of the age group for males and females show statistical significance at the 1% level; they are “age4” for the males and “age1” for the females.

To visualize the association reflected in the p-values of Table 4, we consider a column–tube biplot with interactive coding of age group (column variable) and gender (tube variable). However, since the $Term-IJ$ is statistically significant (see Table 3), we also portray the row–column biplot that depicts the statistically significant association that exists between weight variability and age group ($Term-IJ$). Figure 1 shows these two biplots: on the left side is the column–tube interactive biplot, and on the right side is the row–column interactive biplot. The column–tube interactive biplot on the left of Figure 1 depicts the three-way association where the age group–gender interaction is depicted using principal coordinates, and weight variability is depicted using standard coordinates. The row–column interactive biplot on the right of Figure 1 portrays the three-way association where the weight variability–age group interaction is depicted using principal coordinates, and gender is depicted using standard coordinates. The origin of both plots represents where there is complete independence between the variables. Therefore, the farther the interactive categories are from the origin, the stronger the association. The interactive principal coordinates provide an accurate depiction of their distance from the plot’s origin and from each other. Note that the distance of points depicted by their standard coordinates cannot be appropriately evaluated. Only their angle can be considered: when this angle is acute, there is a strong positive interaction with the interactive pair of categories. When the angle is at 90°, there is independence, and when the angle is obtuse, there is a strong negative interaction.

The quality of the interactive biplots in Figure 1 is very good, since they visually describe about 84% (for the column–tube interactive biplot) and 83% (for the row–column interactive biplot) of the total association. Note that the farthest interactive coordinates contribute most prominently to the association. In the column–tube biplot of Figure 1, they are “age4male”, which is positively related to Q1, and “age1female”, which is positively related to Q4. We can also see from Figure 1 that “age1male” is negatively related to Q1, while “age4female” is negatively related to Q4. In the row–column biplot of Figure 1, we can see that “Q1age4” is positively related to males, while “Q4age1” is positively related to females. Furthermore, “Q1age1” is negatively related to males, and “Q4age4” is negatively related to females.

In summary, the biplots in Figure 1 show that males with T1D and macroalbuminuria are significantly characterized by having “low” weight variability and being in the “old” age group (aged 52 years or older). Figure 1 also shows that the females with T1D and macroalbuminuria are characterized as generally having “high” weight variability and being in the “young” age group (aged 21–27 years old).

6.3. Patients with TD1 and Retinopathy

The primary question we address here is:

What is the strength of the association among weight variability, age group, and gender in patients diagnosed with T1D and retinopathy?

Table 5. Patients with type 1 diabetes experiencing retinopathy, cross-classified by weight variability by age group and gender.

Gender: Male
Age Group
Weight Variability	Age1	Age2	Age3	Age4
Q1	44	56	100	118
Q2	48	67	88	67
Q3	53	66	58	36
Q4	67	58	25	24
Gender: Female
Age Group
Weight Variability	Age1	Age2	Age3	Age4
Q1	36	32	60	97
Q2	38	49	50	66
Q3	62	49	58	40
Q4	58	44	34	35

presents a cross-classification of weight variability (row variable), age group (column variable), and gender (tube variable) among patients experiencing retinopathy, which is the most prevalent microvascular complication experienced by T1D patients, often present during adolescence [32].

Table 6. Partition(s) of Pearson’s three-way chi-squared statistic for patients with T1D and retinopathy of Table 5.

	$\mathit{Term}-\mathit{IJ}$	$\mathit{Term}-\mathit{IK}$	$\mathit{Term}-\mathit{JK}$	$\mathit{Term}-\mathit{IJK}$	$\mathit{Term}-\mathit{Total}$
chi-squared	143.080	9.928	8.052	7.211	168.271
phi-squared	0.080	0.006	0.005	0.004	0.094
% of Inertia	85.029	5.900	4.785	4.286	100.000
df	9.000	3.000	3.000	9.000	24.000
p-value	<0.001	0.019	0.045	0.615	<0.001
Term/df	15.898	3.309	2.684	0.801	7.011

Those p-values in bold are less than the nominal significance level of 0.01.

shows that Pearson’s three-way chi-squared statistic of Table 5 $X_{IJK}^2=168.271$ (df = 24), so that there is a statistically significant association among the three variables (p-value $<0.001$). To assess the nature of the association among weight variability, age group, and gender in detail for patients with T1D and retinopathy, we examine the partition of Pearson’s three-way chi-squared statistic $X_{IJK}^2$ in accordance with Equation (1). This partition is summarized in Table 6, which indicates that the three indices of the bivariate association are statistically significant, albeit at different $\alpha$ levels. The $Term-IJ$ indicates a significant association between weight variability and age group, since the p-value is less than 0.001.

The $Term-IK$, which represents the association between weight variability and gender, shows that there is a gender difference among patients with T1D and retinopathy at the 5% level of significance, and the $Term-JK$ also shows a significant association at this level of significance between age group and gender. However, the trivariate term is not statistically significant.

To determine which categories significantly contribute to the association, we compute the approximate p-value [31] for each category within the weight variability and age group variables of the males and females; see

Table 7. The p-values of categories of the weight variability and age group for males and females with T1D and retinopathy.

p-values of Row Principal Coordinates
Category	p-value
Q1	<0.001
Q2	<0.001
Q3	<0.001
Q4	<0.001
p-values of Column–Tube Principal Coordinates
Category	p-value
age1Male	<0.001
age2Male	0.885
age3Male	<0.001
age4Male	<0.001
age1Female	0.002
age2Female	0.160
age3Female	0.192
age4Female	0.036

Those p-values in bold are less than the nominal significance level of 0.01.

. Table 7 shows that all the categories of weight variability have a p-value that is less than 0.001, showing that their contribution to the association that exists within Table 5 is statistically significant. Additionally, three categories among the age group for males show statistical significance: “age1Male”, “age3Male”, and “age4Male”. However, for the females in the study, only “age1Female” is statistically significant.

Figure 2 illustrates the three-way association. Since the association between age group and gender is statistically significant (see $Term-JK$ in Table 6), this figure only shows the column–tube interactive biplot. Figure 2 shows that the quality of the column–tube interactive biplot is excellent, since 88% of the total association is depicted. In Figure 2, our attention is directed towards the farthest age group–gender points, which are most relevant for evaluating the association. Unlike what was observed for patients with T1D and macroalbuminuria, the biplot shows that males experiencing T1D and retinopathy are mainly characterized by having “high” weight variability and being in the “young” age group (“age1male”). We can also see in Figure 2 that females with T1D and retinopathy are characterized as having a “low” level of weight variability and being in the “old” age group (“age4female”). This result is consistent with what Nyström et al. [32] discussed regarding patients with T1D and retinopathy, observing that overweight/obesity was more common in males than in females.

7. Discussion

The gender gap in patients with T1D exhibits variability across different cardiovascular complications. The analysis of the NDR data reveals that, among the 14 CVCs considered, the association among weight variability, age group, and gender is statistically significant at the $\alpha =0.01$ level of significance for only two complications: macroalbuminuria and retinopathy.

Patients with T1D experiencing macroalbuminuria show distinct patterns. In particular, males are primarily characterized by a “high” level of weight variability (Q4) and being in the “young” age group (“age1male”), while females are characterized by having a “low” weight variability (Q1) and being in the “old” age group (“age4female”).

For those experiencing retinopathy, males are primarily characterized by having a “high” weight variability (Q4) and being in a “young” age group (“age1male”), whereas females are characterized by having “low” weight variability (Q1) and being in the “old” age group (“age4female”), confirming the findings given by Nyström et al. [32].

It is worth noting that, while Pearson’s three-way chi-squared statistic is statistically significant for many of the association terms in Table 3 and Table 6, Table 4 and Table 7 show that the interaction between the categories are not all statistically significant. The computation of p-values in Table 4 and Table 7 helps to assess the significance of the categories, discerning the most relevant to the least.

To visualize the significant associations, we performed three-way CA, an exploratory tool that does not require any distributional assumptions for the variables. This study focused on the gender gap among patients with T1D who experienced a specific CVC, highlighting that a gender gap exists when analyzing the interactions between the diverse categories of weight variability and age class. However, the analysis might be carried out considering other diseases of medical interest to permit personalized treatment, specifically considering gender [33].

Here, the findings also depend on how the weight variability has been classified; we used the CV, which accounts for the differences in averages of the male and female populations. However, other measures of variability might be considered, such as the median absolute difference between subsequent BMI measurements (divided by the interval length starting at the first weight measurement registration), which was used by Nyström et al. [32].