Unlock valuable insights into dairy cattle farming with our statistical analysis exploring the intricate relationship between milk production, lameness, and age. Delve into the data-driven examination of how age influences the association between milk production and lameness, shedding light on key factors affecting dairy cattle health and productivity. Our comprehensive study employs rigorous statistical tests, including the Cochran-Mantel-Haenszel test and odds ratios, to provide a deeper understanding of these critical variables. Discover the significance of age-specific strategies in optimizing milk production and promoting animal welfare in the dairy industry.

## Problem Description

In this statistical analysis assignment, Dairy cattle, bred for their milk production, face a common challenge - lameness, a condition that hinders their movement and welfare. Researchers aim to explore the relationship between milk production and lameness in dairy cows. They believe that age, whether the cow is young or old, may influence this association, thus collecting age-stratified data. Utilizing SAS, we delve into each aspect of this research endeavor.

**Part (a): Odds Ratio and Confidence Interval**

Dairy Cows | Low Milk Production | High Milk Production | Total |
---|---|---|---|

Not Lame | 270 | 160 | 430 |

Lame | 130 | 240 | 370 |

Total | 400 | 400 | 800 |

**Table 1: Cross-Classification of Lameness and Milk Production by Age**

**Odds Ratio:**2.9827**95% Confidence Interval:**(2.0101, 4.4258)

**Interpretation:** Ignoring age, lame cows are approximately 2.98 times more likely to be high milk producers compared to low milk producers, with a significant association indicated by the 95% confidence interval.

**Part (b): Hypothesis Test for Association**

Statistic | DF | Value |
---|---|---|

Chi-Square | 1 | 31.5830 |

Likelihood Ratio Chi-Square | 1 | 31.8468 |

Continuity Adj. Chi-Square | 1 | 30.5312 |

Mantel-Haenszel Chi-Square | 1 | 31.5545 |

**Table 2: Chi-Square and Related Statistics**

**Null Hypothesis:**No association between Milk production and lameness.**Alternative Hypothesis:**An association between Milk production and lameness.

A highly significant p-value of 0.0001 leads to the rejection of the null hypothesis. This highlights a robust association between milk production and lameness, irrespective of age.

**Part (c): Age Modification**

Statistic | Alternative Hypothesis | DF | Value | Prob |
---|---|---|---|---|

1 | Nonzero Correlation | 1 | 37.9189 | .0001 |

2 | Row Mean Scores Differ | 1 | 37.9189 | .0001 |

3 | General Association | 1 | 37.9189 | .0001 |

**Table 3: Cochran-Mantel-Haenszel Statistics**

These statistics confirm that age significantly modifies the association between milk production and lameness in dairy cattle.

**Part (d): Reporting the Mantel-Haenszel OR**

Based on your response in part (c), does it make sense to report the Mantel-Haenszel estimate of the OR? Explain why or why not?

Based on the response in part (c), it does make sense to report the Mantel-Haenszel (MH) estimate of the odds ratio (OR). The CMH test results indicated that age modifies the association between milk production and lameness, which suggests that the relationship between these variables varies across different age groups. As a result, it is essential to consider age as a confounding variable when calculating the association between milk production and lameness.

The Mantel-Haenszel method allows us to calculate the common OR while controlling for age as a stratifying variable. By using the MH estimate of the OR, we can obtain a more accurate representation of the association between milk production and lameness, accounting for age differences. This estimate provides a summary of the relationship that is adjusted for the potential confounding effect of age, making it a more meaningful and robust measure of the true association between these variables in the context of the data.

**Part (e): Mantel-Haenszel Estimate of the Common OR**

Using a stratified analysis, obtain the estimate of the Mantel-Haenszel common OR and it’s 95% confidence interval. Interpret both.

**Common Odds Ratio and Relative Risks**

Statistic | Method | Value | 95% Confidence Limits | |
---|---|---|---|---|

Odds Ratio | Mantel-Haenszel | 3.3832 | 2.2562 | 5.0732 |

Logit | 3.3952 | 2.2687 | 5.0812 |

Table 4: Mantel-Haenszel Estimate of the Common OR

**Interpretation:**After adjusting for age, dairy cows that are lame have approximately 3.38 times higher odds of being in the high milk production group compared to the low milk production group. The 95% confidence interval suggests that we can be 95% confident that the true common odds ratio lies between 2.2562 and 5.0732.

This result indicates a statistically significant and strong association between milk production and lameness in dairy cows, even when age is taken into account. The wider confidence interval reflects some uncertainty due to the stratified analysis, but it still strongly supports the conclusion that lameness is positively associated with high milk production in dairy cows, with a substantial increase in the odds of high milk production in lame cows compared to non-lame cows.

**Part (f): Age as a Confounder**

The concept of age acting as a confounder in the relationship between milk production and lameness in dairy cattle is a critical consideration. A confounder is a variable that is both associated with the exposure (milk production) and the outcome (lameness) and, if not properly controlled for, can distort the observed association.

In this analysis, it becomes evident that age plays a role in influencing the relationship between milk production and lameness. This is substantiated by the change in the odds ratio (OR) when age is adjusted for.

In Part (a), where age is ignored, the OR was estimated at 2.9827. This suggests that lame cows have nearly three times higher odds of being in the high milk production group compared to non-lame cows. However, when we consider age in Part (e) and calculate the Mantel-Haenszel (MH) OR, we obtain a slightly higher OR of 3.3832.

This increase in the OR after age adjustment indicates that age is indeed influencing the association between milk production and lameness. The confounding effect of age is relatively minor, but it's important to recognize its presence. In practical terms, this means that the relationship between milk production and lameness varies across different age groups of dairy cattle.

Age acts as a modifying factor that impacts the strength of the association, albeit to a relatively small degree. As a result, it is essential to account for age as a confounding variable when interpreting the relationship between milk production and lameness in the context of the data. This ensures that the association is accurately understood and provides a more precise representation of the real-world scenario in dairy cattle.

In summary, age is not only an important variable in this analysis but also a potential confounder that needs to be considered in future research and practical interventions aimed at addressing lameness and optimizing milk production in dairy farming. This recognition of age's role as a confounder is crucial for making informed decisions and implementing age-specific strategies for the well-being of dairy cattle.

**Part (g): CMH Test Hypotheses and Validity**

The Cochran-Mantel-Haenszel (CMH) test is a powerful statistical tool used to examine the association between two categorical variables while accounting for a potential confounder, in this case, age. In this part of the analysis, we discuss the hypotheses tested using the CMH test and the validity of the results.

Null Hypothesis and Alternative Hypothesis

The CMH test is a hypothesis test, and as such, it evaluates two competing hypotheses:

**Null Hypothesis:**There is no association between milk production and lameness after controlling for age.**Alternative Hypothesis:**There is an association between milk production and lameness after controlling for age.

The null hypothesis assumes that age has no impact on the relationship between milk production and lameness. It implies that age is not a confounding factor and that the association observed in the data is purely due to chance. On the other hand, the alternative hypothesis suggests that age does influence the association and that the observed relationship is significant and not random.

**Test Statistic and p-value**

The CMH test statistic, calculated based on the stratified analysis of data, provides a measure of the association's strength while adjusting for age. The test statistic in this case is reported as 37.9189.

The p-value associated with this test statistic is crucial for hypothesis testing. In our analysis, the p-value is reported as <0.0001, which means it is extremely small.

Interpretation of p-value: The p-value represents the probability of observing a test statistic as extreme as the one calculated, assuming that the null hypothesis (no association) is true. In this context, the p-value is much smaller than the commonly chosen significance level, typically 0.05.

**Validity of the CMH Test**

The validity of the CMH test is a pivotal consideration in the analysis. In this case, the test is indeed valid for several reasons:

**Appropriate Test for Categorical Data:**The CMH test is designed for the analysis of categorical data, precisely suited for examining the relationship between variables like milk production, lameness, and age.**Controlling for Age:**The primary purpose of the CMH test is to control for age as a potential confounding variable. By stratifying the analysis based on age, it allows us to assess the relationship between milk production and lameness while adjusting for age differences.**Significant Results:**The extremely small p-value (0.0001) is a strong indicator of the validity of the CMH test in this context. It suggests that the association between milk production and lameness, even after considering age, is highly significant and unlikely to be a result of random chance.**Robust Conclusions:**The robustness of the CMH test results reinforces the importance of considering age as a factor in the analysis. It also emphasizes the need to account for age-specific strategies when addressing lameness and optimizing milk production in dairy cattle.

**Part (h): Summary of Findings**

The analysis yields significant insights into the relationship between milk production, lameness, and age in dairy cattle. Ignoring age, a robust association between milk production and lameness is evident. However, the role of age in modifying this association highlights the importance of considering age-specific approaches to address lameness and optimize milk production while ensuring the well-being of dairy cattle.

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