The uncertainty of using carbon footprint as a predictor of overall environmental impacts: comparing linear, logarithmic and circular methods

Correlation and regression analysis

Because correlation and regression are well-established methods, we do not provide a full methodological description and instead refer to standard texts, such as^[18] and^[19]. Nonetheless, a brief recap is warranted, particularly because some previous studies (e.g., ^[8]) conflate correlation and regression, for example, by interpreting R² as a measure of correlation (see^[20] for a detailed discussion).

The sample version of the Pearson correlation coefficient, denoted r, quantifies the strength of the linear association between two numerical variables. It is calculated as

where x_i and y_i are the values of the two variables (e.g., CF and toxicity) for product i, and $$ \bar{x} $$ and $$ \bar{y} $$ denote the sample means, e.g., $$ \bar{x}=\frac{1}{n} \textstyle \sum_{i=1}^{n} x_{i} $$.

The Spearman rank correlation coefficient, ρ. is computed using the ranks of the data. Denoting by x_(i) the rank of x_i within the vector x, one defines

where $$ z=\frac{n+1}{2} $$ is the mean rank. An alternative formula, $$ \rho=1-\frac{6}{n\left(n^{2}-1\right)} \textstyle \sum_{i=1}^{n}\left(x_{(i)}-y_{(i)}\right)^{2} $$, is occasionally used but is less reliable.

Both r and ρ are bounded between [-1,1]. Values of r = 1 and -1 indicate a perfect positive or negative linear relationship, respectively, whereas ρ = 1 or -1 indicates a perfect positive or negative monotonic relationship, which need not be linear. Both coefficients are symmetric (i.e., swapping x and y does not change their value) and are invariant under changes of units or scale, for example, expressing CF in kilograms versus tonnes of CO₂-eq does not affect the results.

Regression analysis, in contrast, assumes a causal relationship between variables. Here, we focus on simple linear regression:

where a and b are the regression coefficients, and e_i is the residual for observation i. The slope b is estimated as

and the constant (or intercept) a by

The quality of the regression fit is typically measured by the coefficient of determination, R²:

R² ranges from 0 to 1, and represents the proportion of variance in y explained by x. A value of R² = 1 indicates perfect prediction of y_i from x_i, whereas R² = 0 indicates no predictive power. Further details on prediction methods are provided in the next section.

For a given regression model, it holds that the coefficient of determination is equal to the squared Pearson correlation coefficient:

However, R² should not be interpreted as a correlation coefficient or “squared correlation”, as is sometimes suggested in the literature. Correlation quantifies the strength of association between two variables in a symmetric manner, without assuming causality. Regression, by contrast, is asymmetric: the estimated coefficients a and b change if the roles of x and y are reversed, reflecting the underlying assumption that one variable predicts or explains the other. Unlike correlation, regression is explicitly intended for predictive purposes. Therefore, while correlation analysis can indicate whether a predictive relationship may exist, it cannot be used to make predictions. Studying the potential of CF as a predictor of other environmental impacts thus requires a regression framework rather than correlation alone.

Prediction with regression analysis

Thus far, the discussion has focused on evaluating the strength of the relationship between CF and other impact indicators across a large sample of products. A more practical question, however, is how to predict the value of a second impact indicator for a specific product when its CF is known. If the goal is to use CF as a predictor, the prediction procedure itself must be explicitly defined.

Perhaps surprisingly, previous studies such as^[4] and^[5] do not address this issue. They report summary statistics, e.g., Pearson’s r, Spearman’s ρ, the coefficient of determination (R²), or other measures of overall association, but these metrics alone do not enable prediction. For example, knowing that a product has a CF of 12 and that the global correlation between CF and toxicity yields R² = 0.78 does not allow one to determine the toxicity score for that product. To make such predictions, a regression model is required, such as

where a and b are estimated regression coefficients and $$ \widehat{y}_{i} $$ is the predicted value of y_i. The difference between predicted and true value is then the residual

Linear and logarithmic relations

When analyzing a sample of products, as in^[4-14], large differences in the magnitude of variables are common. For example, in^[6], the CF axis spans from 10^-12 to 10⁸. On a linear scale, most data points would be compressed into a tiny region, making visualization difficult and potentially affecting correlation and regression analyses. For this reason, logarithmic transformations are often applied. However, it is not always clear whether logarithms are used solely for plotting or also in the statistical calculations. For instance, ^[6] and ^[8] indicate “logarithmic scales” in figure captions but do not clarify whether the analyses were performed on transformed variables.

Applying a logarithm to both variables entails analyzing x’ = log(x) and y’ = log(y). This transformation can substantially affect Pearson’s r, the regression coefficients a amd b, and the coefficient of determination R². In contrast, Spearman’s rank correlation ρ remains unaffected because the logarithm is monotonic. Care is required, however, as logarithms are defined only for positive values. Excluding non-positive data points may alter ρ if these points are removed from the sample.

The base z of the logarithm is not critical. Since $$ \log _{z}(x)=\frac{1}{\ln (z)} \ln (x) $$, changing the base introduces only a multiplicative factor, analogous to converting grams to kilograms. This does not affect Pearson’s r or R², though it will scale the regression coefficients a and b.

Prediction using a logarithmic transformation proceeds as follows. We consider the natural logarithm for illustration. The regression model on the transformed variables is

where x’ = ln(x_i) and y’ = ln(y_i), and a’ and b’ are the regression coefficients estimated from the log-transformed data. The predicted value of y_i on the original scale is then

where a” = exp(a’). This form makes explicit the power-law relationship between the original variables implied by logarithmic regression.

Fallacy of correlation and regression analysis

^[6] analyzed the suitability of using CF as a proxy for multiple impact categories based on a sample of 3,953 products. Their analysis relied on correlation measures, which ^[15] have shown to be vulnerable to methodological limitations. The primary issue is that each data point (x_i,y_i) has an essentially arbitrary scale: it could be rescaled to (cx_i,cy_i) for any c $$ \in \mathbb{R}^{+} $$. For example, one data point may represent 1 kg of concrete, but it could equally represent 1,000 kg, 1 m³, or 1 ounce. Because all impact indicators, including CF, scale proportionally with the quantity of product, rescaling alters the data point’s location and, consequently, the value of the correlation coefficient. Thus, the correlation reported in^[6] depends on 3,953 largely arbitrary choices.

Figure 1 illustrates this issue using a small sample of four points. Three points are fixed (green circles), while the fourth (red square) is rescaled by factors of c = 0.1 in Figure 1B and c = 10 in Figure 1C, producing dramatic changes in the correlation statistics.

Figure 1. (A) Four data points yield a weak correlation; (B) Rescaling the red square data point by a factor of 0.1 produces a strong correlation; (C) Rescaling the same data point by a factor of 10 also results in a strong correlation. The Pearson correlation coefficient is denoted r, and the Spearman rank correlation coefficient is denoted ρ.

Directional statistics

The issue described in the previous section can be addressed by analyzing the ratio between the two variables for each data point:

This ratio is insensitive to changes in scale. If the k-values across the sample are relatively uniform, x serves as a good proxy for y. Conversely, a wide variation in k-values indicates that x is a poor predictor of y. In the example of Figure 1, the k-values are 2.5, 1.75, 2.2, and 5, showing substantial variation, consistent with Figure 1A. This variation, however, is obscured in Figure 1B and C, and is not reflected in the correlation coefficients (r = 0.99, ρ = 1).

Using k-values to calculate averages or standard deviations is not recommended, because the division by x produces highly non-linear behavior when the slope is steep. A more robust approach is to convert the ratio into an angle using the inverse tangent function:

For the data points in Figure 1, the resulting angles are 1.19, 1.05, 1.14, and 1.37 radians. These angles can be visualized by projecting the points onto the unit circle, as illustrated in Figure 2.

Figure 2. The same data points as in Figure 1A-C, rescaled to lie on the unit circle.

These projected points can be analyzed using directional statistics. The average slope is computed as

and the corresponding mean resultant length is

In directional statistics (^[16,17]), the circular variance V is a standard measure of dispersion, providing a robust indicator of the consistency of the slope across the sample:

The circular variance V ranges between 0 and 1. A value of V = 0 indicates that x is a perfect predictor of y, whereas V = 1 corresponds to a complete lack of predictive value. When the analysis is restricted to the first quadrant (i.e., assuming both x and y are non-negative), a uniformly dispersed set of data points yields an average angle $$ \bar{\theta} $$ ≈ 0.79 radians (45°) and a circular variance of approximately V ≈ 0.10. This value can therefore be interpreted as effectively uninformative for predictive purposes: it merely reflects that y tends to be positive when x is positive. Only values of V substantially below 0.10 should be considered indicative of meaningful predictive power. As discussed in the Supplementary Materials, a threshold of approximately V ≈ 0.001 is required to achieve reasonably accurate predictions (see Supplementary Figure 2).

An additional issue concerns the treatment of observations across different quadrants. Consider a dataset consisting of two points, (2,3) and (-2,-3), which lie in the first and third quadrants, respectively. Intuitively, these points should cancel out, yielding a mean resultant length of zero. However, using the single-argument inverse tangent function (ATAN) yields identical angles, since $$ \mathrm{tan}^{-1}(\frac{3}{2} )=\mathrm{tan}^{-1}(\frac{-3}{-2} ) $$ ≈ 0.98 radians (56 degrees). To address this, directional statistics typically employ the two-argument function ATAN2 (instead of the one-argument ATAN), which distinguishes between quadrants. In this formulation, the second point yields an angle of -2.16 radians (-124°), representing the opposite direction, and thus correctly cancels out the first point. While this treatment is appropriate in applications such as wind direction analysis, it is less suitable in the present context. For the purpose of predicting environmental impacts from CF, the sign structure should be interpreted differently: if both CF and the corresponding impact indicator are negative, this situation should be treated analogously to the case where both are positive, as both reflect consistency in direction. Consequently, the use of the ATAN2 function is not appropriate here. It should be noted, however, that ATAN2 still distinguishes between quadrants (1/3 versus 2/4); for example, the points points (2,-3) and (-2,3) both yield an angle of approximately -0.98 radians.

Prediction with directional statistics

The mean angle $$ \bar{\theta} $$ can be used for predictive purposes. Given a value x_i, the corresponding prediction for y_i is obtained as

yielding a predicted value $$ \widehat{y}_{i} $$ for y_i.

Traditionally, prediction error is quantified using the residual e_i = y_i - $$ \widehat{y}_{i} $$. However, this measure is sensitive to the choice of units and scale. For example, predicting y_i = 100 as $$ \widehat{y}_{i} $$ = 150 should be considered equally inaccurate as predicting $$ \widehat{y}_{i} $$ = 50. A ratio such as $$ \widehat{y}_{i} $$/y_i does not satisfy this symmetry. Moreover, predicting y_i = 10 as $$ \widehat{y}_{i} $$ = 15 should be judged equally inaccurate as the previous cases, despite the difference in absolute magnitude.

To address these issues, we define the normalized residual f_i as

This measure is scale-invariant and treats over- and underestimation symmetrically in relative terms. Although introduced in the context of directional statistics, the normalized residual can equally be applied to predictions obtained from conventional regression models.

Fallacies of predicting with regression

An instructive situation arises when the amount of product under analysis is rescaled. Suppose the CF of 1 kg of a product is x, and the corresponding value of another impact category is y. If we instead consider 1,000 kg of the same product, these values scale proportionally to 1000x 1000y. A consistent prediction method should therefore yield a predicted value $$ \hat{y} $$ that also scales proportionally, i.e., from $$ \hat{y} $$ to $$ 1000\hat{y} $$.

To assess this property, we examine the ratio between the prediction for 1,000 units and that for 1 unit. For linear regression, this ratio is

which is generally not equal to 1000, except in the special case where a = 0.

For logarithmic regression, the ratio becomes

which simplifies to 1000^b, and thus equals 1000 only if b = 1.

For the circular method, the ratio is

which is exactly 1000 (provided $$ \tan (\bar{\theta } )\ne 0 $$).

A second relevant case concerns vanishing product quantities. As x → 0, the true impact y also approaches zero. However, under linear regression, $$ \lim_{x \to 0} \hat{y} $$ = a, which is generally non-zero. In the logarithmic case, the model is not defined at x = 0, since ln(0) is undefined. In contrast, the circular method yields $$ \lim_{x \to 0} \hat{y} $$ = 0, which is consistent with the expected behavior.

These observations demonstrate that prediction of one impact indicator from another, specifically, predicting environmental impacts from CF, is structurally more consistent with directional statistics than with conventional regression approaches.

Data

We used the dataset of 3,953 products originally analyzed in ^[6], which includes values for the carbon footprint as well as a range of additional environmental impact categories (see Table 1 for an overview).

For each impact category, we computed the Pearson correlation coefficient (r) for both the original data and log-transformed data, the Spearman rank correlation coefficient (ρ), and the directional statistic, expressed as the circular variance (V).

To evaluate predictive performance, we selected a set of representative products spanning different categories:

• 1-pentanol, at plant (1 kg, region RER)

• aluminium, production mix, at plant (1 kg, region RER)

• barge tanker (1 piece, region RER)

• electricity mix (1 MJ, region DE)

• operation, average train (1 person-km, region DE)

• potatoes, at farm (1 kg, region US)

For each of these six products, the acidification impact was estimated using three prediction methods: linear regression, logarithmic regression, and directional statistics. The predicted values were then compared to the observed values using the normalized residual as a measure of prediction error.

Overall quality of fit

The main results are summarized in Table 1.

The results reveal several inconsistencies across metrics. For example, acidification exhibits strong correlation statistics, yet performs among the worst when evaluated using directional statistics. Conversely, land use shows a relatively low Pearson correlation coefficient but ranks among the best according to circular variance. It should be noted that the linear and logarithmic Pearson coefficients, as well as the non-parametric Spearman coefficient, are derived from methods that we have argued to be theoretically inappropriate for this research question. The discrepancies observed here therefore underscore that our critique is not merely theoretical but also has clear empirical implications.

To facilitate interpretation, selected results are visualized. Figure 3 presents the case of ozone depletion. On a linear scale [Figure 3A], a clear positive association is apparent, with higher values of x corresponding to higher values of y. On a logarithmic scale [Figure 3B], the association remains visible, although a number of data points deviate from the general pattern. The circular representation [Figure 3C] provides additional insight. While the first quadrant (positive x and y) is densely populated, the points are widely dispersed within this quadrant, without a clear concentration. The resulting circular variance is 0.02, indicating poor suitability as a proxy. In addition, the fourth quadrant contains a non-negligible number of observations: approximately 10% of products exhibit a negative CF while maintaining non-negative ozone depletion scores, further weakening predictive performance.

Figure 3. Scatterplots of the carbon footprint (horizontal axis) versus ozone depletion impact (vertical axis) for the product sample. Panel (A) shows both axes on a linear scale; panel (B) uses logarithmic scales; panel (C) presents the data projected onto the unit circle (“proj.”).

Although ozone depletion represents one of the better-performing impact categories, aquatic eutrophication [EP (P)] is among the worst. The corresponding results are shown in Figure 4.

Figure 4. Scatterplots of the carbon footprint (horizontal axis) versus aquatic eutrophication EP (P) impact (vertical axis) for the product sample. Panel (A) shows both axes on a linear scale; panel (B) uses logarithmic scales; panel (C) presents the data projected onto the unit circle (“proj.”).

In the original analysis by ^[6], the Spearman correlation coefficient was used, yielding values “between 0.7 and 0.8 for all impact categories except land use”, which “suggests a good correlation between CF ... and the rest of the environmental impact scores”. ^[6] further reports that the Spearman correlation coefficients are associated with “P < 0.05”. This likely refers to a hypothesis test of the null hypothesis of no correlation (ρ = 0), which is rejected at the 95% confidence level. While this may appear compelling, such significance is largely driven by the large sample size. With n = 3,953, even a very weak correlation (e.g., ρ ≈ 0.03) will be deemed statistically significant. As a result, statistical significance in this context provides limited information about the practical strength or relevance of the association]. In contrast, the directional statistical analysis presented here leads to a markedly different conclusion. All circular variance values are 0.02 or higher, and the patterns observed in Figures 3C and 4C reveal substantial dispersion.

In a subsequent refinement, ^[6] qualifies its initial conclusions by noting that energy-related products exhibit lower correlations (“below 0.7”), whereas infrastructure-related products show higher correlations (“above 0.94 for all impact categories”). This prompted us to revisit the case of infrastructure products using directional statistics. The results are shown in Figure 5.

Figure 5. Scatterplots of the carbon footprint (horizontal axis) versus aquatic eutrophication EP (P) impact (vertical axis) for 488 infrastructure-related products. Panel (A) shows both axes on a linear scale; panel (B) uses logarithmic scales; panel (C) presents the data projected onto the unit circle (“proj.”).

While the Spearman correlation remains high (0.96), the directional analysis provides a more nuanced perspective. Although most observations lie in the first quadrant, they remain widely dispersed within it, resulting in a circular variance of 0.05. This value exceeds the proposed threshold of 0.001 by a factor of 50, indicating poor predictive performance despite the high correlation coefficient.

Predicted values

Table 2 presents the acidification scores alongside their predicted values obtained using three methods: linear regression, logarithmic regression, and directional statistics, for six representative products (see Section 2.8 for details).

Table 3 reports the corresponding normalized (relative) residuals for these predictions.

Across these illustrative cases, linear regression performs poorly. The logarithmic and circular approaches yield comparatively better results, although their relative performance varies by product. For instance, for pentanol, directional statistics overestimates the acidification impact by only 3%, whereas for aluminium, the logarithmic regression provides a near-perfect estimate. However, for electricity, all three methods produce substantial errors, with deviations of a factor of two or more.

Potatoes provide an additional instructive example. Due to their negative CF, logarithmic regression cannot be applied. Moreover, the circular method produces an estimate with the incorrect sign (negative instead of positive). This highlights a broader issue: products with negative CF values (often arising from carbon fixation) pose challenges for prediction. In this case, the negative CF likely reflects only the production phase; a full cradle-to-grave assessment would likely yield a positive value. Such cases may therefore need to be excluded from predictive analyses.

It is important to emphasize that Tables 2 and 3 are not intended to determine which method performs best. Rather, our objective is to demonstrate that all three approaches exhibit limited predictive accuracy. The key distinction lies in interpretation: while linear and logarithmic regression produce high goodness-of-fit metrics (with R² exceeding 0.95), these metrics give a misleading impression of predictive quality. In contrast, circular variance provides a more realistic assessment by reflecting the underlying dispersion.

Figure 6 presents scatterplots of predicted versus observed acidification values for all 3,953 products. The dashed line represents perfect prediction.

Figure 6. Scatterplots of true (horizontal axis) versus predicted (vertical axis) acidification impacts using three methods. Panel (A) shows the full dataset; panels (B-D) provide zoomed views of regions with smaller values.

Because the true acidification values span approximately 18 orders of magnitude, Figure 6A compresses much of the detail, necessitating the zoomed representations in Figure 6B-D.

Figure 7A shows the distribution of normalized residuals for acidification predictions. From a predictive perspective, the most relevant interval is f $$ \in $$ [-10%,10%], corresponding to errors within ±10%. Linear regression performs very poorly within this range. Logarithmic and circular methods achieve this level of accuracy for 11.6% and 6.3% of observations, respectively. Expanding the tolerance to [-50%,50%], these proportions increase to 50.9% for logarithmic regression and 41.5% for the circular method. Consequently, the probability that the acidification score of a randomly selected product is misestimated by more than 50% lies between 50% and 60%, indicating substantial predictive uncertainty.

Figure 7. Histograms of normalized residuals for the full sample using three prediction methods: (A) acidification; (B) freshwater ecotoxicity.

Similar patterns are observed for other impact categories. Figure 7B, which presents results for freshwater ecotoxicity, exhibits the same general structure: extremely poor performance for linear regression and comparatively weak, but broadly similar, performance for the logarithmic and circular methods.