### INTRODUCTION

### MATERIALS AND METHODS

### Data

### Statistical analysis

*y*

*is the milk yield of the*

_{ijkl}*l*th animal of the

*i*th herd-year-season, recorded at the

*j*th calving age and the

*k*th DIM;

*HYS*

*is the effect of the*

_{i}*i*th herd-year-season (

*i*= 1 to 2,483);

*age*

*is the effect of the*

_{j}*j*th calving age (

*j*= 1 to 2);

*DIM*

*is the effect of the*

_{k}*k*th DIM (

*k*= 1 to 50);

*a*

*and*

_{l}*p*

*are the effects of the additive genetic merit and permanent environment of the cow l (*

_{l}*l*= 1 to 56,132);

*f*(

*i*) is the heat stress function for the herd-test-day (HTD);

*v*

*and*

_{l}*q*

*are the additive and permanent environmental effects of heat tolerance on cow l, respectively; and*

_{l}*e*

*is the residual effect.*

_{ijkl}*f*(

*i*) was calculated as:

### RESULTS AND DISCUSSION

*r*= 0.87–0.97) to the 305-day evaluation in a previous study [13]. Although a sophisticated test-day model, i.e., a random regression model of the lactation curve, can capture heat stress variation throughout the lactation period, this method has much greater difficulty in estimating the parameters [7]. Therefore, the simple test-day model is more appropriate than the sophisticated test-day model.

*r*

*) was −0.33, indicating that the selecting animals for high milk yield could produce animals susceptible to heat stress. According to a previous study, the total body heat load of lactating cows increased according to the metabolic heat production associated with milk production [14], in turn increasing the ability to maintain homeothermy under heat stress conditions [15]. Other studies have reported correlation coefficients of −0.33 [7] and −0.38 [5], and other similar values [16,17].*

_{a,v}