# Concentration–Response Functions as an Essence of the Results from Lags

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Health Data

_{3}) concentration was considered. Its levels were estimated and represented as an 8 h daily maximum concentration. Daily average ambient temperature and relative humidity were applied to represent these weather values. The weather parameters in the constructed models were included in the form of natural spines. The environmental data were retrieved from the National Air Pollution Surveillance (NAPS) database, maintained by Environment and Climate Change Canada [12].

#### 2.2. Statistical Model

_{3}, where ambient ozone is measured as a maximum of averaged concentrations over an 8 h period. The constructed models are realized by implementing a conditional Poisson regression. The conditional Poisson model is conditioning on the total event count in each stratum. The time stratified case-crossover (CC) technique is applied to determine the strata, which are composed of 4 or 5 days [13,14]. The constructed strata match days are based on a hierarchical calendar structure, which is as follows: same day of the week, calendar month, and year. These constructions have previously been used to minimize bias among various approaches to define control periods for the case event in the case-crossover models [13]. This construction can eliminate time-invariant factors such as sex, smoking, and socioeconomic position. It has been shown that a conditional Poisson regression model gives equivalent estimates as the conditional logistic regression model [15]. The models realized here allow adjusting for over-dispersion (using the option: quasi-Poisson). In this case, there is no AIC value.

#### 2.3. Transformation

- No transformation; $AP=T\left(Z\right)=Z,\mathrm{and}RR\left(Z\right)=exp\left(Beta\times Z\right)$. It is a classical approach on the log-linear scale. Here, it is called the CC method.
- The transformation has the form $T\left(Z\right)=f\left(Z\right)\times LWF\left(Z\right)$, where f(Z) is a simple function of Z, such as Z, log(Z), sqrt(Z) (= $\sqrt{Z})$, or some other powers of Z ($f\left(Z\right)={Z}^{P}),$ with P greater or lower than 1, i.e., convex and concave functions of the variable Z, respectively.
- The transformation has the following form $T\left(Z\right)=f\left(Z\right)\times LWF\left(Z\right)$, with the function $f\left(Z\right)=log\left(1+Z/A\right)$, where A is a parameter. This form is also used to represent a pooled C-RF shape.

## 3. Results

## 4. Discussion

_{3}concentration. Another is their profile and shape. The estimated values of RR have almost the same range for the three proposed approaches (a–c). The main differences are in their shape. The methods, which have a better fit to the data, using the goodness of fit criterion, give the lowest estimation at the point of 30 ppb (see Table 3, AIC values). Moreover, this method suggests the presence of a threshold.

^{3}) were RR = 1.01947 (95%CI: 1.00295, 1.03627) using the standard case-crossover model, and RR = 1.00391 (95%CI: 1.00181, 1.00601) using the case-crossover model with the transformation. The corresponding AIC values were 78,707.7 vs. 78,699.3 [22].

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Daily values during the study period (3652 days). Histogram of ED visits (daily counts), ozone (ppb), temperature (°C), and relative humidity (%).

**Figure 2.**

**Left panel**: C-RF shapes by lags.

**Right panel**: a pooled C-RF shape. No transformation $\left(f\left(Z\right)=Z\right)$.

**Figure 3.**

**Left panel**: C-RF shapes by lags.

**Right panel**: a pooled C-RF shape. Transformation with f(Z) chosen among three simple functions.

**Figure 4.**

**Left panel**: C-RF shapes by lags.

**Right panel**: a pooled C-RF shape. Transformation with $f\left(Z\right)=log\left(1+Z/A\right)$.

**Table 1.**The estimations obtained by the case-crossover approach: (

**a**) no transformation, (

**b**) transformation with f(Z) (=Z, log(Z), or sqrt(Z)), and (

**c**) transformation with f(Z) = log(1 + Z/A). The transformation is submitted into the models in the form f(Z) × LWF(Z) to represent concentrations. ED visits for all respiratory conditions. Edmonton, Canada, 1992–2002.

(a) RR(Z) = exp(Beta × Z) | |||||
---|---|---|---|---|---|

Lag | Beta | SE | Lag | Beta | SE |

0 | 0.00183 | 0.00039 | 5 | 0.00131 | 0.00039 |

1 | 0.00115 | 0.00039 | 6 | 0.00151 | 0.00039 |

2 | 0.00104 | 0.00039 | 7 | 0.00171 | 0.00039 |

3 | 0.00184 | 0.00039 | 8 | 0.00144 | 0.00039 |

4 | 0.00195 | 0.00039 | 9 | −0.00004 | 0.00039 |

(b) RR(Z) = exp(β(Z)), β(Z) = Beta × f(Z) × LWF(Z,τ,μ), τ = 0.1 | |||||

Lag | Beta | SE |
%
| μ | f(Z) |

0 | 0.0099 | 0.0019 | 39 | 15.0 | sqrt |

1 | 0.0069 | 0.0020 | 34 | 13.8 | sqrt |

2 | 0.0010 | 0.0003 | 31 | 13.0 | z |

3 | 0.0165 | 0.0032 | 26 | 11.6 | log |

4 | 0.0182 | 0.0032 | 26 | 11.6 | log |

5 | 0.0146 | 0.0033 | 60 | 20.6 | log |

6 | 0.0087 | 0.0019 | 57 | 19.6 | sqrt |

7 | 0.0095 | 0.0020 | 27 | 12.0 | sqrt |

8 | 0.0014 | 0.0003 | 49 | 17.6 | z |

9 | −0.0041 | 0.0041 | 0 | 1.2 | log |

(c) RR(Z) = exp(β(Z)), β(Z) = Beta × log(1 + Z/A) × LWF(Z,τ,μ) | |||||

Lag | Beta | SE | μ | τ | A |

0 | 0.0467 | 0.0070 | 14.0 | 0.005 | 15.2 |

1 | 0.0035 | 0.0006 | 14.0 | 0.005 | 0.1 |

2 | 0.1220 | 0.0325 | 13.8 | 0.005 | 106.5 |

3 | 0.0047 | 0.0007 | 13.9 | 0.005 | 0.4 |

4 | 0.0414 | 0.0064 | 10.8 | 0.005 | 11.0 |

5 | 7.3226 | 1.4687 | 16.0 | 0.005 | 5690.1 |

6 | 4.3945 | 0.9644 | 17.4 | 0.061 | 3250.2 |

7 | 0.0650 | 0.0118 | 10.9 | 0.005 | 28.0 |

8 | 326.6490 | 77.3902 | 17.4 | 0.081 | 243,940.0 |

9 | −0.0085 | 0.0018 | 3.1 | 0.005 | 0.0 |

**Table 2.**Estimations based on the RR values calculated for lag from 0 to 8. Three forms of the input (

**a**)–(

**c**) are realized. ED visits for all respiratory conditions. Edmonton, Canada, 1992–2002.

Parameters | Value | SE | Low Value | SE | Upper Value | SE |
---|---|---|---|---|---|---|

Output | (a) Input: RR(Z) = exp(Beta × Z) | |||||

θ | 0.4 | 4.5 | 2.9 | 5052.0 | 2.3 | 1715.4 |

A | 130.7 | 1805.4 | 3275.0 | 6,745,000.0 | 748.9 | 745,095.2 |

μ | −27.6 | 655.0 | −1158.0 | 822,200.0 | −388.6 | 415,055.2 |

τ | 2.3 | 27.6 | 12.9 | 7879.0 | 7.0 | 1336.5 |

Output | (b) Input: RR(Z) = exp(β(Z)),β = Beta × f(Z) × LWF(Z,τ,μ) | |||||

θ | 0.0313 | 0.0027 | 0.0164 | 0.0020 | 0.0465 | 0.0034 |

A | 8.1101 | 1.4794 | 6.9103 | 1.8306 | 8.6736 | 1.3260 |

μ | 14.2728 | 0.3625 | 14.2329 | 0.5102 | 14.2752 | 0.3073 |

τ | 0.1190 | 0.0020 | 0.1188 | 0.0027 | 0.1190 | 0.0018 |

Output | (c) Input: RR(Z) = exp(β(Z)), β(Z) = θ × log(1 + Z/A) × LF(Z,τ,μ) | |||||

θ | 0.0816 | 0.0047 | 0.0420 | 0.0025 | 0.1240 | 0.0073 |

A | 50.9900 | 3.6630 | 39.6000 | 3.0700 | 57.8600 | 4.1180 |

μ | 13.0700 | 0.0491 | 13.0100 | 0.0567 | 13.1000 | 0.0469 |

τ | 0.0274 | 0.0009 | 0.0267 | 0.0010 | 0.0277 | 0.0008 |

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Szyszkowicz, M.
Concentration–Response Functions as an Essence of the Results from Lags. *Int. J. Environ. Res. Public Health* **2022**, *19*, 8116.
https://doi.org/10.3390/ijerph19138116

**AMA Style**

Szyszkowicz M.
Concentration–Response Functions as an Essence of the Results from Lags. *International Journal of Environmental Research and Public Health*. 2022; 19(13):8116.
https://doi.org/10.3390/ijerph19138116

**Chicago/Turabian Style**

Szyszkowicz, Mieczysław.
2022. "Concentration–Response Functions as an Essence of the Results from Lags" *International Journal of Environmental Research and Public Health* 19, no. 13: 8116.
https://doi.org/10.3390/ijerph19138116