### 4.1 Average crash probability

Prior to the analysis of the statistical models, the average probability that at least one road crash occurs within one hour in a German administrative district is computed for each of the 78 crash types considered (Fig. 1). If all crashes are considered without distinguishing between specific crash characteristics, the hourly probability is 9.487%. Probabilities are lower if computed for more specific vehicle types. For example, the probability for single-car or a single-truck crashes is 1.441% and 0.111%, respectively. The lower crash probability for trucks can at least partly be attributed to a lower number of trucks on the roads and to a lower vulnerability of trucks due to their structural characteristics.

Crashes are further classified in terms of the speed limit at the crash location, crash type, road environment and crash severity. For certain crash types probabilities are relatively low, in particular in case of some sub-types of single-truck crashes (hit-object crashes, crashes at crossroads and with fatalities), where the probability is 0.002. This should be kept in mind when interpreting the RRI values for these crash types.

### 4.2 Functional relationships

Within the frame of this paper a detailed discussion of models for all 78 crash types is infeasible. Instead, as an example, we discuss the functional relationships between the occurrence probability of multi-car rear-end crashes and the different predictor variables. For this discussion, we focus on one predictor term in Eq. 5 at a time. Modeled crash probability is then plotted against this term, while keeping the predictor variables of the other terms constant at selected values.

An increase in traffic volume leads to a non-linear but monotonous increase in rear-end crash probability, if all other parameters are held constant (Fig. 2a). This is a reasonable behavior, which indicates that the spatially aggregated traffic data can adequately represent the effects of traffic volume on crash probability at district level.

To account for different basic properties in different administrative districts, such as the total number of vehicles per district, we have included the average hourly crash probability \(\bar{p}_{a,d}\) in the model. In districts with larger \(\bar{p}_{a,d}\) the model shows that also the crash probability \(p_a\) under specific meteorological conditions is larger (see Fig. 2b). The relationship between \(\bar{p}_{a,d}\) and \(p_a\) is approximately piecewise linear with a change in slope at around \(\bar{p}_{a,d}=0.1\). Note that values of \(\bar{p}_{a,d} >0.1\) are only reached in a few highly populated districts.

Rear-end crash probability shows a small variability in time after controlling for traffic volume and meteorological factors (Fig. 2c). A systematic trend in rear-end crash probability is not evident. However, it should be noted that other crash types show a decreasing trend, which could be attributed to advanced safety features of cars, for example (not shown).

An increase in wind speed leads only to a small increase in probability of rear-end crashes (Fig. 2d). If the confidence intervals are taken into account, this increase is not significant. This is not surprising, since one would not expect a large impact of wind speed on this crash type, but the following section will reveal wind speed impacts on other crash types.

There is a clear impact of summer precipitation (\({\text {Tmp}}=15^\circ\)C) on rear-end crash probability (indicated by color and line type in Fig. 2a and b). If the district average hourly precipitation changes from 0 to 1 mm/h, crash probability increases by a factor of 1.548, which corresponds to an RRI of 54.8%. A further increase in precipitation to 2 and 3 mm/h leads to smaller increases in crash probability, but it is still significant with respect to the confidence intervals.

At negative temperatures the increase in rear-end crash probability is even stronger in case of increasing precipitation, which is shown in a visualization of the combined effect (interaction) of precipitation and surface temperature (Fig. 2e). For example, if precipitation changes from 0 mm/h to 1 mm/h at -3\(^\circ\)C, crash probability increases by 80.1%. The varying effect of precipitation depending on temperature can be described by including the two variables as an interaction term in the generalized additive model. One could expect a sharper increase in crash probability around 0\(^\circ\)C, however, the functional relationship shows a rather smooth transition between positive and negative temperatures. This can be attributed to different sources of uncertainty, which are represented by the smoothing term. For example, the actual road surface temperatures at specific locations within a district can differ from the aggregated surface temperatures based on the gridded ERA5 data used in the model.

Finally, the visualization of the combined effect of cloud cover and sun elevation angle shows that there are two areas with increased probabilities of rear-end crashes (Fig. 2f). First, increased probabilities occur at positive sun elevation angles combined with low cloud cover. This could be attributed to a distraction of drivers due to sun glare, leading to a larger number of rear-end crashes. Second, an increasing probability is also observed with increasingly negative elevation angles, indicating that the sun is below the horizon. This increase could be attributed to reduced visibility under low-light conditions during night time. Note that probabilities are computed assuming a constant traffic volume. At night times one can expect traffic volume (and thus also crash probability) to decrease, which counteracts the increase in probability due to low light conditions.

### 4.3 Increase in crash probability due to weather conditions

#### 4.3.1 Winter precipitation

For the 78 crash types, we compute the *Relative Risk Increase* (RRI) for hours with winter precipitation with respect to hours without winter precipitation as described in the methods section. Under the selected conditions (Table 3), precipitation is most likely snowfall or freezing rain. In general, single-vehicle crashes (i.e. single-car and single-truck crashes) show a larger RRI compared to multi-vehicle crashes (i.e. multi-car crashes and crashes including all vehicle types; Fig. 3). Single-truck crashes show the largest RRI of 872.9%.

A higher (or even no) speed limit at the location of the crash leads to a larger RRI under conditions with winter precipitation. In case of single-truck crashes the RRI increases to 1521.7% at speed limits of 130 km/h. However, it should be noted that generally the maximum speed of trucks above 3.5 t is limited to 80 km/h.

Run-off-road and head-on crashes also show a relatively large RRI under conditions with winter precipitation, indicating that vehicles tend to leave their lane due to slippery road conditions. The RRI for other crash types are smaller, but in most cases positive and significant. When comparing different road environments, RRI values are largest in curves and descending road segments. Single-truck crashes also show a strong increase on ascending roads segments. If the RRI under winter precipitation conditions is computed separately for crashes with different severities, it is evident that less severe crashes show larger RRI values compared to more severe crashes.

#### 4.3.2 Summer precipitation

Analogously to winter precipitation, we investigate the effect of summer precipitation (most likely rainfall). The RRI is computed for hours with summer precipitation with respect to hours without summer precipitation. In case of summer precipitation, crash probability increases for all crash types (Fig. 4). However, the increase is generally smaller than in case of winter precipitation. Similar to winter precipitation, summer precipitation leads to larger RRI values in case of single-vehicle crashes, at higher driving speeds, as well as in case of run-off-road crashes and in curves, descents and ascents. Probabilities of less severe crashes increase more than those of severer crashes. While in case of winter precipitation RRI values for single-truck crashes were generally larger than single-car crashes, in case of summer precipitation the opposite is true. The largest RRI of 536.2% is found in case of single-car crashes at speed limits of 130 km/h and above.

#### 4.3.3 Low sun elevation at cloud-free conditions

To evaluate the effect of sun glare on crash probability, we compute the RRI for hours with low sun elevation angle and cloud-free conditions with respect to hours with low sun elevation angle and full cloud cover. The RRI of different crash types under low sun and cloud-free conditions range from -35% to +52% (Fig. 5). In case of multi-vehicle crashes, probabilities increase for most crash types. The largest RRI values of multi-car crashes crashes occur at speed limits of 130 km/h and more (\(+52.5\)%) and in case of rear-end crashes (\(+43.2\)%). These increases could be attributable to sun glare, which could lead to reduced visibility and increased reaction times of drivers, which is particularly dangerous at high driving speeds and in dense traffic.

In case of single-car and single-truck crashes, RRI values are negative in most cases. However, in case of single-trucks these decreases are mostly not significant. These decreases of crash probability of single-vehicle crashes under low sun and cloud-free conditions could be due to the fact that we have not taken time lagged effects of precipitation into the design of our models. If within the hour of the crash there was no rain, the road surface is more likely to be dry under cloud-free conditions (e.g. due to evaporation effects due to sunshine) and more likely to be wet under cloudy conditions (e.g. due to possible precipitation at previous time steps). A higher likelihood for a dry road (with higher surface friction) consequently leads to the observed reductions of single-car crash probabilities, which is particularly large in case of curves (-35.3%).

#### 4.3.4 Extreme wind speeds

To evaluate the effect of extreme wind speeds on crash probability, the RRI at hours with high wind speeds is computed with respect to hours with low wind speeds (Fig. 6). In general, the RRI values in case of extreme wind speeds are relatively small, compared to the effects of the other meteorological parameters analyzed above, and mostly not significant, except for single-truck crashes. Those show a significant RRI of 104.9%, which is in line with other studies showing that trucks are particularly vulnerable to high wind speeds [12]. RRI values of single-truck crashes are largest at high speed limits between 100 and 130 km/h. Since the maximum speed of trucks is limited to 80 km/h, this effect could be explained by the assumption that highways with such high speed limits often run through open rural terrain and are particularly exposed to high wind speeds.

What stands out are the hit-object crashes, which increase in probability by 413.6% for single-car crashes and by 789.8% for single-truck crashes. These crashes can be attributed to crashes with broken tree branches, debris, or other objects, which are blown onto the road by strong winds.

### 4.4 Cross-validation results

The predictive power of the model \(\mathcal {M}_{\text {met}}\) and whether meteorological terms in the predictor improve the predictions compared to a model without these (\(\mathcal {M}_{\text {nomet}}\)) is analyzed in a two-fold cross-validation experiment using the AUC and AUCSS. The AUC values for the 78 different crash types mainly range between 0.7 and 0.85. Values between 0.7 and 0.8 correspond to an acceptable discrimitation, values above 0.8 correspond to an excellent discrimination [20].

For all crash types (except for single-truck crashes with fatalities), the AUCSS values are positive, indicating an improvement of the models ability to discriminate between time steps with and without crashes due to the meteorological predictor variables (Fig. 7). The AUCSS ranges between relatively low values of 0.44% in case of single-car crashes with fatalities to high values of 24.21% in case of single-car crashes at locations with high speed limits of 130 km/h and above. In general, AUCSS values are higher in case of those crash types, which also showed a strong relationship to one or more of the meteorological variables. Highest AUCSS values occur in case of single-vehicle crashes, at higher speed limits, at locations with curves, descents or ascents and in case of crashes without or with minor injuries. Lower AUCSS values occur in case of multi-car crashes at lower speed limits, in case of crashes with other vehicles (except head-on crashes), at crossroads and in case of crashes with severe injuries and fatalities.